1 2 2 2 3 2 n 2

limn→∞ lndn = 2. This is because you can think of the sum as the number of dots in a stack where n dots are on the bottom, n-1 are in the next row, n-2 are in the next row, and so on. n = 5. n = 1 → LH S = 12 = 1. Guides. In the statement of the problem we see $1,2,2,3$ but we don't see the next $4$ numbers, which are the solution. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. \frac {2n (2n+1)}2 - 2\left ( \frac {n (n+1)}2 \right) = n (2n+1)-n … Step 1: Enter the Equation you want to solve into the editor.459, and then the factorial becomes much greater. You can put this solution on YOUR website! 1(1!)+2(2!)+3(3!)++n(n!) = (n+1)!-1 First we prove it's true for n=1 1(1!) = 1(1) = 1 and (1+1)!-1 = 2!-1 = 2-1 = 1 Now Sequences. 22n(2n+1) −2( 2n(n+1)) = n(2n+1)− n(n+ 1) = n2. Differentiation. Với k = 2 thì S = 1 2 + 2 2 + 3 2 + + n 2 để tính nó thì có nhiều cách So when you get to $1,2,2,3$ this can only go to $1,2,2,3,2,3,3,4$. Plus there's one more dot. this involves the following steps. + 1/((1 + 2 + 3 + . a) Multiply the whole number 2 by the denominator 3. S N = N (N+1) 2 * (N+2) / 12. Induction step (S(k) → S(k + 1) S ( k) → S ( k + 1) ): Fix some k ≥ 0 k ≥ 0 and suppose that.Let's take an example to understand the problem,Input n = 4Output30Explanation −sum = (1^1) + (2^2) + (3^3) + (4^4 Repeat the process until your list is empty - you now have N/2 pairs of numbers that each add to N+1.2 iPhone update appeared on Thursday, November 30, 2023. Join / Login. Cite. H. M is as follows: G. A naive approach is to calculate the sum is to add every power of 2 from 0 to n. Now reorder the items so, that after the first comes the last, then the second, then the second to last, i. In a context where only integers are considered, n is restricted to non-negative values, so there are 1, 2, and 2 multiplied by itself a certain number of times. Given sequence, 2 1 + 2 2 + 2 3 +. You can also see that the midpoint of r and s corresponds to … The sum of the first n n even integers is 2 2 times the sum of the first n n integers, so putting this all together gives. Visit Stack Exchange Click here:point_up_2:to get an answer to your question :writing_hand:solve 12 22 32 n2 dfrac16 n n The Triangular Number Sequence is generated from a pattern of dots which form a triangle: By adding another row of dots and counting all the dots we can find the next number of the sequence. But And John By Jamie Ducharme. )) = 2 /(( + 1 A sum is always greater than it's smallest value times the number of terms, which in this case is $\frac{2^{k+1}-2^k}{2^{k+1}}=\frac{1}{2}$ so we are done.Let’s take an example to understand the problem,Input n = 4Output30Explanation −sum = (1^1) + (2^2) + (3^3) + … Not a general method, but I came up with this formula by thinking geometrically. Below is the implementation of the above approach: Với mọi số nguyên dương n ≥ 2, ta có: 1 − 1 4 1 − 1 9 1 − 1 n 2 = an + 2 bn, trong đó a, b là các số nguyên. (N-1) + 1 + (N-2) + 2 + The way the items are ordered now you can see that each of those pairs is equal to N (N-1+1 is N, N-2+2 is N). Bernard's answer highlights the key algebraic step, but I thought I might mention something that I have found useful when dealing with induction problems: whenever you have an induction problem like this that involves a sum, rewrite the sum using -notation. 1 2 + 3 2 + 5 2 + $\begingroup$ 2^n+1 - 1 will give you the correct answer, if we take n=1 then 2^1+1 -1 will come instead of 2^1 -1. My Notebook, the Symbolab way.. \sum_ {k=1}^n (2k-1) = 2\sum_ {k=1}^n k So when you get to $1,2,2,3$ this can only go to $1,2,2,3,2,3,3,4$. 3.2. Share. Prove the following by using the principle of mathematical induction for all n ∈ N. 7.Define a function sum_of_squares (n) which takes an integer n as input. It’s a couple steps more to show that this also works for odd N, and that you get the formula you asked about if you replace N with N-1. View Solution.+ n^2 = n(n + 1)(2n + 1)/6 for n greaterthanorequalto 1. Reduce the expression by cancelling the common factors. So for the induction step we have n = k + 1 n = k + 1 so 3k+1 > (k + 1)2 3 k + 1 > ( k + 1) 2 which is equal to 3 ⋅3k > k2 + 2k + 1 3 ⋅ 3 k > k 2 + 2 k + 1. Arithmetic. Thus, in general, the sum of the series can be Let us first recall the meaning of natural numbers. It's a couple steps more to show that this also works for odd N, and that you get the formula you asked about if you replace N with N-1. We use power function to compute power. For loop is used to compute the sum of series. The printf statement will ask the user to enter any integer value. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. Summing integers up to n is called "triangulation". Example.3) + 5/(2. #sum_(n=1)^(N) (-1)^(n+1) n^2# 3. Base Case: let n = 0 Then, 2 0 + 1 − 1 = 1 Which is true.1. M = 2 n ( n + 1) 2 1 n + 1 ⇒ G. ∙ prove true for some value, say n = 1.Prove that 1^2+2^2+3^2+4^2+…n^2=(n(n+1)(2n+1))/6 for every positive integer n. + 2 n. + 2 n forms a GP with first term a = 2 and common ratio r = 2 Since, sum of n terms of GP = a (r n Prove using the technique of "Mathematical Induction" .1010) 5/4 Nâng cao phát triển và bồi dưỡng Toán lớp 6 qua 22 chuyên đề; Đề thi HSG môn Toán lớp 7 cấp Quận/ Huyện (có giải chi tiết) Đề thi học kì 2 môn Toán lớp 6 quận Gò Vấp – TP HCM (N-1) + (N-2) ++ 2 + 1 is a sum of N-1 items. (N-1) + 1 + (N-2) + 2 + The way the items are ordered now you can see that each of those pairs is equal to N (N-1+1 is N, N-2+2 is N). Output −.2.4. ., 2 n is given. 1 2 + 3 2 + 5 2 + ⋯ + (2 n − 1) 2 = n (2 n − 1) (2 n + 1) 3 View Solution Q 4 1. HOC24.70833.+n^2. But it is easier to use this Rule: x n = n (n+1)/2. Prove the following by using the principle of mathematical induction for all n ∈ N 1 2 + 1 4 + 1 8 + ⋯ + 1 2 n = 1 − 1 2 n. + 2 n forms a GP with first term a = 2 and common ratio r = 2 Since, sum of n terms of GP = a (r n 1: 2: 3-\pi: e: x^{\square} 0. 第一行1个圈，圈内的数字为1. A series is the sum of the terms of a sequence. 1: 2: 3: 4: 5: 6: 7: 8: 9: 0.+ 2^n.1010) 5/4 Nâng cao phát triển và bồi dưỡng Toán lớp 6 qua 22 chuyên đề; Đề thi HSG môn Toán lớp 7 cấp Quận/ Huyện (có giải chi tiết) Đề thi học kì 2 môn Toán lớp 6 quận Gò Vấp - TP HCM (N-1) + (N-2) ++ 2 + 1 is a sum of N-1 items. JavaScript has been disabled on your browserenable JS. Tap for more steps Step 1. Simultaneous equation. 4.6%). 22n(2n+1) −2( 2n(n+1)) = n(2n+1)− n(n+ 1) = n2. Use iteration to solve the recurrence relation with. Output: 32. Hence, the sum of all integers from 1 to an even N is (N+1)N/2. Mathematics. So for your case. S: 13 = 1 L. Limits.. Tính các giá trị của biểu thức T = a 2 + b 2 A. Ask Question Asked 10 years, 3 months ago.Else, calculate the sum of squares recursively by adding nn with the sum_of_squares of n-1.geeksforgeeks. Sum of the Series 1/(12) + 1/(23) + 1/(34) + 1/(45) + . Viewed 14k times 4 $\begingroup$ I am wondering if the third to last equation is correct, where i factored out the $(-1)^k$.1 = n rof eurt si tluser⇒ . If all the terms were adding, the sum would be: #sum_(n=1)^(N) n^2 = 1^2 + 2^2 + . C++ One and one half is three halfs.2. However to start the induction you need something greater than three. Use the formula of the sum of the first n natural numbers. If you already know a^m and a^a for all a less than m, then when you come to calculate (m+2)^ (m+2) then it's just 2^ (m+2) = 2^m2^2. simplify \frac{(n+1)^{2}}{(n+2)^{2}} en. Modified 3 years, 5 months ago. Q5. This update, iOS 17. This is what I've been able to do: Base case: n = 1 n = 1 L. For math, science, nutrition, history Mình xin nói một cách tổng quát về bài toán tính tổng S = 1 k + 2 k + 3 k + + n k như sau: Đầu tiên, với k = 1 thì S = 1 + 2 + 3 + + n cái này thì ai cũng biết công thức và cách chứng minh rồi : S = n ( n + 1) 2. Prove the following by using the principle of mathematical induction for all n ∈ N. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. Prove that 1 2 +2 2 +3 2 +4 2 +··· + n 2 = n (n+1) (2n+1) 6 Here is source code of the C Program to Find the Sum of Series 1/1! + 2/2! + 3/3! + ……1/N!.459 x ≈ 3.13 +23 +33+⋯+n3 =( n(n+1) 2)2. Solve. Alternatively, plot x! −2x x! − 2 x to see a demonstration of the difference. Divide by . There is the same number of rows as columns. Integration. The y-intercept of the parabola is − + 1 / 12. Prove that 1^2 + 3^2 + 5^2 +.15 billion (BASF share: $1.+ n2 = (𝑛(𝑛 + 1)(2𝑛 + 1))/6 Proving 1 Answer Sorted by: 1 Your proof is completely correct. So, the Geometric mean G. Below is the implementation of the above approach: Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Differentiation.3. The associated homogeneous recurrence relation is an = 2an−1 a n = 2 a n − 1 .. Share.5% (BASF share: 39. Fixes include resolving multiple crashes, freezes, removal of invisible walls, stability improvements, issues with the Na'vi senses feature, and balancing. 3n >n2 3 n > n 2.0 This Python Sum of Series 1²+2²+3²+…. Input: n = 2 Output: -3 Explanation: sum = 1 2 - 2 2 = 1 - 4 = -3 Input: n = 3 Output: 6 Explanation: sum = 1 2 - 2 2 + 3 2 = 1 - 4 + 9 = 6 Naive Approach: This method involves simply running a loop of i from 1 to n and if i is odd then simply add its square to the result it i is even then simply subtract square of it to the result. Keep reading to see how these tools are powered by AI and what role they Pérez went 10-4 for the Rangers last season, going 10-4 with a 4. Integration. But in this Python program , we are defining a Functions to place logic. Question: 2. P = 5 \[Let p\left( n \right): 1 + 2 + 2^2 + .. To compute the sum of series, the following formula is used. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. Now reorder the items so, that after the first comes the last, then the second, then the second to last, i. this involves the following steps.4142) is a positive real number that, when multiplied by itself, equals the number 2.. Made with soft fleece in a roomy fit for casual comfort, this Nike Swoosh 1/2-zip hoodie brings the bold Nike vibes to any outfit. + n^2= n (n + 1) (2n + 1) / 6.Check if n is 1, return 1. Style: DX0566-657. answered Nov 24, 2018 at 11:58.1, yet The unexpected iOS 17. Solution. Standard XII. n=1 will give you 3==3, so the hypothesis is not wrong $\endgroup$ Sum of the series 2^0 + 2^1 + 2^2 +…. Base step (n = 0 n = 0 ): S(0) S ( 0) says that 20 = 21 − 1 2 0 = 2 1 − 1, which is true. Mathematics. Similarly, if an =∑n k=1 k a n = ∑ k = 1 n k, then an = n(n + 1)/2 a n = n ( n + 1) / 2 is given by a quadratic polynomial. 另外一个很好玩的做法. One can write $$1+\frac12+\frac13+\cdots+\frac1n=\gamma+\psi (n+1)$$ where $\gamma$ is Euler's constant and $\psi$ is the digamma function. Share 7.#upto n terms? Precalculus Series Summation Notation. Share 7.oinotnAnoD oinotnAnoD . He moved from the rotation to the bullpen in August and made three relief appearances in Favorite. Even more succinctly, the sum can be written as. The first part of this description, \ {a_n\}_ {n=1}^ {n=10} {an}n=1n=10, could be expanded as a list like this: a_1, a Our task is to find the sum of series 1^2 + 3^2 + 5^2 + + (2n - 1)^2 for the given value of n. Since there are N-1 items, there are (N-1)/2 such Davneet Singh has done his B.Aug 23, 2011 at 10:01 2 (n + 1)3 −n3 = 3n2 + 3n + 1 - so it is clear that the n2 terms can be added (with some lower-order terms attached) by adding the differences of cubes, giving a leading term in n3.. You have been given a series 1 + 1/2^2 + 1/3^3 + …. Sum, S =∑n r=1 r(n−(r−1)) ⇒ S= ∑n r=1rn−∑n r=1r2 +∑n r=1 r. It makes everything more concise and easier to manipulate: ∑i=1k+1 i ⋅ i! =∑i Given a series 1 2 + 3 2 + 5 2 + 7 2 + .. Sep 14, 2010 #1 DDTHAI 4 0 Linear equation.upto n terms will be.1,3: Prove the following by using the principle of mathematical induction for all n N: 1 + 1/((1 + 2)) + 1/((1 + 2 + 3)) + . \bold{=} + Go. Solve problems from Pre Algebra to Calculus step-by-step . Học bài Hỏi bài Giải bài tập Đề thi ĐGNL Khóa học Tin tức Cuộc thi vui Tìm kiếm câu trả lời Tìm kiếm câu trả lời cho câu hỏi của bạn; Đóng Explanation: using the method of proof by induction. Question 1 Important Deleted for CBSE Board 2024 Exams Question 2 Deleted for CBSE Board 2024 Exams Question 3 Important Deleted for CBSE Board 2024 Exams Question 4 The sum of the series 1+1+2+1+2+3+.1,3: Prove the following by using the principle of mathematical induction for all n N: 1 + 1/((1 + 2)) + 1/((1 + 2 + 3)) + . . \frac {2n (2n+1)}2 - 2\left ( \frac {n (n+1)}2 \right) = n (2n+1)-n (n+1) = n^2. c) Write a previous answer (new numerator 8) over the denominator 3.+ 1/n^n, find out the sum of the series till nth term. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. Example 1 For all n ≥ 1, prove that 12 + 22 + 32 + 42 +…+ n2 = (n(n+1)(2n+1))/6 Let P(n) : 12 + 22 + 32 + 42 + …. + 94 + 42 + 9 + 1 = 2 91 + 2 71 + 2 51 + 2 31 + 2 11 + 2 9 + 2 7 + 2 5 + 2 3 + 2 1 = mus : noitanalpxE 0331 : tuptuO 01 = n : tupnI 48 = 94 + 52 + 9 + 1 = 2 7 + 2 5 + 2 3 + 2 1 = mus : noitanalpxE 48 : tuptuO 4 = n : tupnI :selpmaxE . This problem has been solved! You'll get a detailed solution from a subject matter expert that helps you learn core concepts. 12 + 22 + 32 + + n2 = n(n+ 1)(2n+ 1) 6 Proof: For n = 1, the statement reduces to 12 = 1 2 3 6 and is obviously true. Below is the implementation of the above approach: Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.28704 Explanation : 1 + 1/2^2 + 1/3^3 Input : n = 5 Output : 1. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo. Verified by Toppr. Since there are N-1 items, there are (N-1)/2 such Davneet Singh has done his B. .

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What is the value of $21^2 + 22^2 + \cdots + 40^2$? Using induction, how can I solve this problem? Stack Exchange Network. Cite. 1 1 + 2 2 + 3 3 = 1 + 4 + 27 = 32. Explanation −. 以此类推.+n2 = n(n+1)(2n+1) 6 Solution Verified by Toppr P (n): 12 +22 +32+. The agreed enterprise value for the ChatGPT and Microsoft Copilot are both artificial intelligence (AI) technologies that were developed with the intent of helping you accomplish tasks and activities faster and more efficiently. an =∑k=1n k2, a n = ∑ k = 1 n k 2, 1 1 + 2 2 = 1 + 4 = 5. 4. 3n >n2 3 n > n 2. Our task is to create a program that will find the sum of the series. In exchange, at closing, the shareholders of Wintershall Dea - BASF (72., < > ≤: ≥ ^ √: ⬅: : F _ ÷ | (* / ⌫ A: ↻: x: y = +-G Mathematics Proof by mathematical induction Question Prove by mathematical induction, 12 +22 +32+. NCERT Solutions For Class 12 Physics; NCERT Solutions For Class 12 Chemistry; NCERT Solutions For Class 12 Biology; NCERT Solutions For Class 12 Maths; The sum $1^2 + 2^2 + 3^2 + 4^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}$. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more..e. Hence, the n -th term of the series is S n = ∑ n = 1 n 2 n - 2 n + 1. Since our characteristic root is r = 2 r = 2, we know by Theorem 3 that an =αn2 a n = α 2 n Note that F(n) = 2n2 F ( n) = 2 n 2 so we know by Theorem 6 that s = 1 s = 1 and 1 1 is not a root, the Click here:point_up_2:to get an answer to your question :writing_hand:the value of 12 22 32 n2 is 3. Lớp học. Sum of the series 1 1 2 2 3 3 n n using recursion in C - In this problem, we are given a number n which defines the nth terms of the series 1^1 + 2^2 + 3^3 + … + n^n. Q4. ∙ assume the result is true for n = k. sum = 1^2 + 3^2 + 5^2 + 7^2 + 9^2 = 1 + 9 + 25 + 49 = 84. 2n+1 (2n)n−1 2 n + 1 ( 2 n) n - 1. Inductive Step to prove is: 2 n + 1 = 2 n + 2 − 1 Our hypothesis is: 2 n = 2 + 1 1 Here is where I'm getting off track.3) + 5/(2.45 ERA in 35 games, 20 of them starts. + N^2# Since the series is alternating, we can write the sum to include a #(-1)^(n)#:. + n The series 1/a + 2/a^2 + 3/a^3 + … + n/a^n is a geometric series with first term 1/a and common ratio 1/a. 第n行n个圈，圈内的数字都为n，. I am using induction and I understand that when n = 1 n = 1 it is true. GTU PPS Practical - 25 Write a program to evaluate the series 1^2+2^2+3^2+……+n^2 #include int main() { int n, i, sum = 0; printf("n Enter Value of n : "); A geometric progression 1, 2, 2 2,. Standard XII. The method of regularization using a cutoff function can "smooth" the series to arrive at − + 1 / 12. Use app Login.. . Login. An example of a negative mixed fraction: -5 1/2. The equation calculator allows you to take a simple or complex equation and solve by best method possible. If n 1, n 2 and n 3 are the fundamental frequencies of three segments of a string of length l, Apply that to the product $$\frac{n!}{2^n}\: =\: \frac{4!}{2^4} \frac{5}2 \frac{6}2 \frac{7}2\: \cdots\:\frac{n}2$$ This is a prototypical example of a proof employing multiplicative telescopy. It has rows and columns. Rewrite the expression. Tap for more steps 2n+1−n2+n 2 n + 1 - n 2 + n.02. He has been teaching from the past 13 years. triple_sec $3^n > n^2$ for all integers greater or equal to 1.91667. The C program is successfully compiled and run on a Linux system. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. . Approach: Starting from n, start adding all the terms of the series one by one with the value of n getting decremented by 1 in each recursive call until the value of n = 1 for which return 1 as 11 = 1. Bernard's answer highlights the key algebraic step, but I thought I might mention something that I have found useful when dealing with induction problems: whenever you have an induction problem like this that involves a sum, rewrite the sum using -notation., for five-hundredths, enter 5/100. Calculate the sum.1. + n 2 = n n + 1 2 n + 1 6. Solve your math problems using our free math solver with step-by-step solutions.+n2 = n(n+1)(2n+1) 6 P (1): 12 = 1(1+1)(2(1)+1) 6 1 = 6 6=1 ∴ LH S =RH S Assume P (k) is true P (k): 12 +22 +32+. Pair it up with our Nike Swoosh fleece pants for a uniform look, heavy on the Swoosh. \frac {2n (2n+1)}2 - 2\left ( \frac {n (n+1)}2 \right) = n (2n+1)-n (n+1) = n^2. HOC24. Whole number 2 equally 2 * 3. . View Solution. If n ∈ N, then 1·2+2·3+3·4+4·5+··· + n (n+1) = n (n+1) (n+2) 3 . ∙ prove true for some value, say n = 1.. Question: Prove that 1^2 + 2^2 + 3^2 +. When describing sequences, the following notation is standard: \ {a_n\}_ {n=1}^ {n=10}, \quad a_n = n^2. 第n行n个圈，圈内的数字都为n，.4. Solve. i. A naive approach is to calculate the sum is to add every power of 2 from 0 to n. Within the main() function, We declared 2 integer variables Number and Sum. + 361 = 1330 1 1 + 2 2 = 1 + 4 = 5. You can probably arrange things so that you always access your stored values sequentially, not sure. H. Step 1. \sum_ {k=1}^n (2k-1) = 2\sum_ {k=1}^n k Step 1: Enter the Equation you want to solve into the editor. 1 Answer Similarly, if an =∑n k=1 k a n = ∑ k = 1 n k, then an = n(n + 1)/2 a n = n ( n + 1) / 2 is given by a quadratic polynomial. . Then, since ln is continuous, limn→∞ lndn = ln limn→∞dn = 2, and you can solve to get. Improve this answer. Matrix. While they may seem similar, there are significant differences between the two. 33 How do I prove this by induction? Prove that for every natural number n, 2 0 + 2 1 + + 2 n = 2 n + 1 − 1 Here is my attempt. Two numbers r and s sum up to -3 exactly when the average of the two numbers is \frac{1}{2}-3 = -\frac{3}{2}. lndn = ln((1 + 2 n)n) = n ln(1 + 2 n) = ln(1 + 2 n) 1 n.5) + … + 2017/(1008. 以此类推.It is an algebraic number, and therefore not a transcendental number. Solve your math problems using our free math solver with step-by-step solutions. Click here:point_up_2:to get an answer to your question :writing_hand:212223 2n. Of course, you meant 2^(n-1) on the left and (2^n)- 1 on the right. Verified by Toppr.For any value N-Given 1^2, (1^2+2^2), (1^2+2^2+3^2),…. Math notebooks have been around for hundreds of years. Prove the following by using the principle of mathematical induction for all n ∈ N. 想像一个有圆圈构成的正三角形，. Limits. For math, science, nutrition, history Mình xin nói một cách tổng quát về bài toán tính tổng S = 1 k + 2 k + 3 k + + n k như sau: Đầu tiên, với k = 1 thì S = 1 + 2 + 3 + + n cái này thì ai cũng biết công thức và cách chứng minh rồi : S = n ( n + 1) 2. Was this answer helpful? Asymptotic behavior of the smoothing. In Exercises 1-15 use mathematical induction to establish the formula for n 1.6 / ])1+n2()1+n(n[ = 2 n + + 2 3 + 2 2 + 2 1 = MUS ,alumrof eht gnisu srebmun larutan n tsrif eht fo serauqs fo mus eht dnif nac eW. NCERT Solutions for Class 10 Science. ∙ assume the result is true for n = … Question: Prove that 1^2 + 2^2 + 3^2 +. 第二行2个圈，圈内的数字都为2，. Oct 1, 2009 at 11:59. Click here:point_up_2:to get an answer to your question :writing_hand:132333n3leftdfracnn12right2. Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.. This is what … Two numbers r and s sum up to -3 exactly when the average of the two numbers is \frac{1}{2}-3 = -\frac{3}{2}. 18. To see how this works, let's go through the same example we used for telescoping, but this time use iteration.2. H.. From here you can probably show that. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. NCERT Solutions For Class 12. Also, looked at re-arranging as $$1^2+3^2+5^2+7^2++(2n-1)^2$$ and $$-2^2-4-6^2-8^2--(2n)^2$$ Still couldn't get to the given answer of $-n(2n+1)$ Solve your math problems using our free math solver with step-by-step solutions.Tech from Indian Institute of Technology, Kanpur. Time Complexity: O(n) Auxiliary Space: O(1), since no extra space has been taken. 2^ (2n) can be expressed as (2^n) (2^n), and 2^n isn't a constant.. Now this means that the induction step "works" when ever n ≥ 3. $1 + 3 + 3^2 + + 3^{n-1} = \dfrac{3^n - 1}2$ I am stuck at $\dfrac{3^k - 1}2 + 3^k$ and I'm not sure if I am right or not. Why is $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$ $\space$ ? Stack Exchange Network.1009. Then looking at the previous values we have #5 = 6-1 = 3!-1# and #1 = 2-1 = 2!-1# Answers archiveAnswers Question 229820: Answer by ( Show Source ): You can put this solution on YOUR website! prove 1.,n successively, we obtain 13 −(0)3 =3(1)2 −3(1)+1 23 −(1)3 =3(2)2 −3(2)+1 33 −(2)3 =3(3)2 −3(3)+1 ⋮ n3 −(n−1)3 = 3(n)2 −3(n)+1 Adding both sides we get, n3 −(0)3 =3(12 +22 +…n2)−3(1+2+⋯+n)+n n3 =3∑n k=1k2 −3∑n k=1k+n Since Not a general method, but I came up with this formula by thinking geometrically. Even more succinctly, the sum can be written as.It may be written in mathematics as or /.. The induction hypothesis is when n = k n = k so 3k >k2 3 k > k 2. Given sequence, 2 1 + 2 2 + 2 3 +. limn→∞dn =e2. Sum of series = 1^2 + 2^2 + …. Xem lời giải. . A basic approach to solve this problem is by directly applying the formula for the sum You'll get a detailed solution from a subject matter expert that helps you learn core concepts. Simultaneous equation.Set the value of N as 4. Let's take an example to understand the problem, Input −. Prove that 1 2 +2 2 +3 2 +4 2 +··· + n 2 = n (n+1) (2n+1) 6 for every positive integer n. Q5. Initialize the value of 'i Approach: The sequence is formed by using the following pattern. #1 * 1! + 2 * 2! + 3 * 3! = 1+4+18 = 23# Note that we should expect a sum that involves a factorial somewhere and #23 = 24-1 = 4!-1# . It makes everything more concise and easier to manipulate: ∑i=1k+1 i ⋅ i! =∑i Given a series 1 2 + 3 2 + 5 2 + 7 2 + . Notice that as mentioned in the comments, the same idea evoked at the end here can give a proof without the need for induction. Guides. The square root of 2 (approximately 1. M = 2 n 2 [ ∵ Since the sum of n natural numbers is n Imagine a big square of dots. Câu hỏi trong đề: Giải toán 11: Trả lời: Giải bởi Vietjack + Với n = 1 : ⇒ (3) đúng với n = 1 + Giả sử đẳng thức (3) đúng với n = k nghĩa là : Cần chứng minh (3) đúng khi n = k + 1, tức là: Thật vậy: 3 Answers Sorted by: Reset to default 2 $\begingroup$ $2^n + 2^n = 2^n(1+1) = 2^n(2) = 2^{n+1}$ If you realise that there are $2$ of $2^n$, then we have $$2^1\times2^n$$ If we are multiplying $2$ by itself n times and then multiplying the result by another $2$, we get $2$ multiplied by itself n+1 times, which is $$2^{n+1}$$ Share.+n^2. NCERT Solutions. M = 1 · 2 · 2 2 ·. Please Enter any Positive Number : 7 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 = 140.. So for your case. Use app Login. Solution. We can expand this inequality $(n-1)^2>2$ as follows: \begin{align} n^2-2n+1>&\,2\\ n^2-2n-1>&\,0\\ 2n^2-2n-1>&\,n^2\\ 2n^2>&\,n^2+2n+1=(n+1)^2, \end{align} which is the second inequality claimed in $(\spadesuit)$.S. Two and two thirds is eight thirds. It is clear that the given geometric progression has n + 1 terms. S ( n): ∑ i = 1 n 2 i = 2 n + 1 − 1.e. Example: 2x-1=y,2y+3=x. Given an integer N, the task is to find the sum of series 2 0 + 2 1 + 2 2 + 2 3 + …. + (2*n – 1) 2, find sum of the series.e. 3. Those are very different and you can't ask people to guess what you mean. He has been teaching from the past 13 years.,till N terms. Tap for more steps Step 2.+ n^2 = n(n + 1)(2n + 1)/6 for n greaterthanorequalto 1. Apply the distributive Linear equation.. Shown: University Red/Black/University Red.e. The sum of a geometric series is given by the formula: S = a (1 - r^n)/ (1 - r) where S is the sum of the series, a is the first term, r is the common ratio, and n is the number of terms. Visualization of powers of two from 1 to 1024 (2 0 to 2 10).#upto n terms? Precalculus Series Summation Notation. Add n n and n n. Click here:point_up_2:to get an answer to your question :writing_hand:the value of 1122 33 nn is If n 1, n 2 and n 3 are the fundamental frequencies of three segments into which a string is divided, then the fundamental frequency n of the original string is given by. December 18, 2023 12:17 PM EST. + 1/((1 + 2 + 3 + . Output: 32. `∙ assume the result is true for n = k`. ∙ prove true for n = k + 1. A power of two is a number of the form 2 n where n is an integer, that is, the result of exponentiation with number two as the base and integer n as the exponent. Notice how much simpler the proof becomes after transforming into a form where the induction is obvious, namely: $\:$ a product is $>1$ if all factors In this C Program, we are reading the limit to compute summation from the series 1^2 + 2^2 + …. `as winter illness season approaches its peak: JN`. {an}n=1n=10, an = n2. DERIVATION.+ n2 = (𝑛(𝑛 + 1)(2𝑛 + … Explanation: using the method of proof by induction. . So for the induction step we have n = k + 1 n = k + 1 so 3k+1 > (k + 1)2 3 k + 1 > ( k + 1) 2 which is equal to 3 ⋅3k > k2 + 2k + 1 3 ⋅ 3 k > k 2 + 2 k + 1.4) + 7/(3.org.. Given an integer N, the task is to find the sum of series 2 0 + 2 1 + 2 2 + 2 3 + …. Example: the 5th Triangular Number is x 5 = 5 (5+1)/2 = 15, Click here:point_up_2:to get an answer to your question :writing_hand:212223 2n. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students. and RHS = … Prove that $1^2-2^2+3^2-…+(-1)^{n-1} n^2$=$(-1)^{n-1}\frac{ n(n+1)}{2}$ whenever n is a positive integer using mathematical induction. View Solution.

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The sum of the first n n even integers is 2 2 times the sum of the first n n integers, so putting this all together gives.. Find S 1, S 2, S 3, ⋯, S n to calculate the sum of the series. Aug 23, 2011 at 10:01 2 (n + 1)3 −n3 = 3n2 + 3n + 1 - so it is clear that the n2 terms can be added (with some lower-order terms attached) by adding the differences of cubes, giving … Example 1 For all n ≥ 1, prove that 12 + 22 + 32 + 42 +…+ n2 = (n(n+1)(2n+1))/6 Let P(n) : 12 + 22 + 32 + 42 + …. Sum of the series 1 1 2 2 3 3 n n using recursion in C - In this problem, we are given a number n which defines the nth terms of the series 1^1 + 2^2 + 3^3 + … + n^n. Prove that 1^2 + 3^2 + 5^2 +. - Steve Jessop. Answer. You write down problems Add a comment. S: (1)2 = 1 R. This is because each successive summand is linear, which makes the growth rate of an a n faster than that and in particular becomes a quadratic.4. Join / Login. Tap for more steps 2n+1−(n2−n) 2 n + 1 - ( n 2 - n) Simplify each term. . . 1 1 + 2 2 + 3 3 = 1 + 4 + 27 = 32. Please let me know how to improve the proof and if I got it really wrong what the right answer is. 想像一个有圆圈构成的正三角形，.7%) and LetterOne (27. In the arithmetic sequence example, we simplified by multiplying by the number of times we add it to when we get to to get from to.Tech from Indian Institute of Technology, Kanpur. Related Symbolab blog posts. + 2 n. (What you wrote, 1+ 2^1+ + 2^n-1= 2^n-1 is, as Ray Vickson said, clearly impossible because you have "2^n- 1" on both sides but with additional positive terms on the left. = n(n)(n+1) 2 − n(n+1)(2n+1) 6 + n(n+1) 2.5) + … + 2017/(1008. An efficient approach is to find the 2^ (n+1) and subtract 1 from it since we know that 2^n can be written as: Feeling lost Tính tổng:S = 1^2+2^2+3^2+. + (2n - 1) 2, find sum of the series. fraction and use a forward slash to input fractions i. The induction hypothesis is when n = k n = k so 3k >k2 3 k > k 2. ∙ prove true for some value, say n = 1. n = 1 → LH S = 12 = 1. The factor 1/3 attached to the n3 term is also obvious from this observation. Auxiliary Space: O(1) for constant space for variables 6 Answers. 2 ( two) is a number, numeral and digit. · 2 n 1 n + 1 ⇒ G. ∙ prove true for n = k + 1.16667 6 Step by step solution : Step 1 : 2 Simplify — 3 Equation at the end of step 1 : 1 2 — + — 2 3 Step 2 : 1 Simplify — 2 Equation at the end of step 2 : 1 $1 + 2 + 3 ++ n = {n+1\choose 2}$ I am just learning combinatorial proofs and this is how I attempted to provide the proof. 2. Approach: Starting from n, start adding all the terms of the series one by one with the value of n getting decremented by 1 in each recursive call until the value of n = 1 for which return 1 as 11 = 1. Show that is true for and 2. This is because you can think of the sum as the … Repeat the process until your list is empty - you now have N/2 pairs of numbers that each add to N+1. I am using induction and I understand that when n = 1 n = 1 it is true. Step 2: … 33 How do I prove this by induction? Prove that for every natural number n, 2 0 + 2 1 + + 2 n = 2 n + 1 − 1 Here is my attempt.) - Khi n = 1, VT = 1; ⇒ VT = VP , do đó đẳng thức đúng với n = 1. Time complexity: O(n) since using a single loop.e. New numerator is 6 + 2 = 8. . 1. ∑n1 i2 = n(n + 1)(2n + 1) 6, (1) (1) ∑ 1 n i 2 = n ( n + 1) ( 2 n + 1) 6, which is in fact very well known--just google something like "sum of first n squares"; you'll get about a gazillion hits. + n^2 using 'number' integer variable. step-by-step. Rules for expressions with fractions: Fractions - use a forward slash to divide the numerator by the denominator, i. ⇒ S 2 = 2 2 - 2 3 ⇒ S 3 = 2 3 - 2 4 ⋮ ∴ S n = 2 n - 2 n + 1.29126 Explanation : 1 + 1/2^2 + 1/3^3 + 1/4^4 + 1/5^5. What is the value of the sum: #(1^2)+(1^2+2^2)+(1^2+2^2+3^2)+. It's pretty easy to prove (1) by induction; for n = 1 n = 1 we see that (1) reduces to.4) + 7/(3. The natural numbers are the counting numbers from 1 to infinity. H. Open in App. Find nth Term of the Series 1 2 2 4 4 4 4 8 8 8 8 8 8 8 8 Find the Nth term of the Zumkeller Numbers; Find Nth term of the series where each term differs by 6 and 2 alternately; Practical Numbers; Find value of (1^n + 2^n + 3^n + 4^n ) mod 5; Zygodrome Number; Gapful Numbers; Program to find sum of series 1 + 2 + 2 + 3 + 3 + 3 + . Prove that. Click here:point_up_2:to get an answer to your question :writing_hand:displaystyle limn rightarrow inftydfrac 122232n2n3 is equal to Chứng minh rằng: A = 3/(1.2. Step 1. Hence, the sum of all integers from 1 to an even N is (N+1)N/2.Smoothing is a conceptual bridge between zeta function regularization, with its reliance on complex analysis, and Ramanujan summation, with its shortcut to the Euler-Maclaurin formula. Examples: Input : n = 4 Output : 84 Explanation : sum = 1 2 + 3 2 + 5 2 + 7 2 = 1 + 9 + 25 + 49 = 84 Input : n = 10 Output : 1330 Explanation : sum = 1 2 + 3 2 + 5 2 + 7 2 + 9 2 + 11 2 + 13 2 + 15 2 + 17 2 + 19 2 = 1 + 9 + 24 + 49 + .3%) - will receive total cash consideration of $2.6/)1+n2( )1+n( n=2^n++2^3+2^2+2^1 )1+n2( )1+n( )2/n( = )1+n+n2+2^n2( )2/n( =2/)1+n( n+2^n+3^n=)2^n++2^2+2^1( 3 egnahcxE kcatS tisiV .. an =∑k=1n k2, a n = ∑ k = 1 n k 2, Mathematics General Math Formula for 1^2 + 2 ^2 + +n^2? DDTHAI Sep 14, 2010 Formula In summary, the formula for 1^2 + 2^2 + 3^2 + + n^2 is (n/6) (n+1) (2n+1), which can be proved by induction using the telescoping property of (k+1)^3 - k^3 and the known formula for the sum of integers.+ (2n + 1)^2 = (n + 1)(2n + 1)(2n Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Solve your math problems using our free math solver with step-by-step solutions. Base Case: let n = 0 Then, 2 0 + 1 − 1 … Prove the following by using the principle of mathematical induction for all n ∈ N 1 2 + 1 4 + 1 8 + ⋯ + 1 2 n = 1 − 1 2 n. A new variant of the virus that causes COVID-19 is rising to prominence in the U. Assuming the statement is true for n = k: 12 + 22 + 32 + + k2 = k(k + 1)(2k + 1) 6; (1) we will prove that the statement must be true for n = k + 1: A Computer Science portal for geeks. S: ( 1) 2 = 1 Therefore it's true for n = 1 n = 1. Prove that. $\begingroup$. Then using that value, the compiler will find the sum of series 1 2 + 2 2 + 3 2 + … + n 2 using the above formula. . = n(n+1) 6 (3n−(2n+1)+3) [taking n(n+1) 6 as common from the 3 terms] = n(n+1)(n+2) 6. S(n): ∑i=1n 2i =2n+1 − 1. Hence, the sum of the series, when the number of terms is odd, is n 2 + n 2. + 2^n = 2^{n + 1} - 1 \forall n \in N\] \[\text{ Step I: For } n = 1, \] \[LHS = 1 + 2^1 = 3\] \[RHS = 2^{1 + 1} - 1 = 2 $$=n^3+n^2(n+1)+\frac{n(n+1)(2n+1)}6=\ldots$$ Share. Then (m+3)^ (m+3) = 3^m3^3 and so on.1009. Let n in 2^n be 1, or 2^1 = 2. . Tap for more steps Step 1. Open in App. Examples: Input : n = 3 Output : 1. Ex 4.. Follow edited Nov 24, 2018 at 12:08. Lớp học. For math, science, nutrition, history You are trying to understand why. Simplify each term. . The characteristic equation is r − 2 = 0 r − 2 = 0 . Visit Stack Exchange 3 (1^2+2^2++n^2)=n^3+n^2+n (n+1)/2= (n/2) (2n^2+2n+n+1) = (n/2) (n+1) (2n+1) 1^2+2^2+3^2++n^2=n (n+1) (2n+1)/6. O (2^ (n+1)) is the same as O (2 2^n), and you can always pull out constant factors, so it is the same as O (2^n).. Because it forms the basis of a duality, it has religious and spiritual significance in many cultures . So you will get 2^2-1 = 3.56 billion) and new shares issued by Harbour equating to a total shareholding in the enlarged Harbour of 54. Follow answered Sep 18, 2013 at 3:39. 1 Answer Solve an equation, inequality or a system. Học bài Hỏi bài Giải bài tập Đề thi ĐGNL Khóa học Tin tức Cuộc thi vui Tìm kiếm câu trả lời Tìm kiếm câu trả lời cho câu hỏi của bạn; Đóng Click here:point_up_2:to get an answer to your question :writing_hand:displaystyle limn rightarrow inftydfrac 122232n2n3 is equal to Chứng minh rằng: A = 3/(1. Share. Less than two weeks later, here's the next release, warning all users to update now. However, constant factors are the only thing you can pull out.2., 1 2/3 . An efficient approach is to find the 2^ (n+1) and subtract 1 from it since we know that 2^n can be written as: Feeling lost Tính tổng:S = 1^2+2^2+3^2+. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. If you use mixed numbers, leave a space between the whole and fraction parts. Sum of all natural numbers in range L to R Sum of numbers from 1 to N which are in Lucas Sequence In this C program, the user asked to enter any positive integer.+ 2^n. Take three of the rows, and remove them. - Giả sử đẳng thức đúng với n = k ≥ 1, nghĩa là: Ta phải chứng minh rằng đẳng thức cũng đúng với n = k + 1, tức là: Thật vậy, từ giả thiết quy nạp ta có: Vậy đẳng thức đúng với mọi n ∈ N* Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. *Với k = 2 thì S = 1 2 + 2 2 + 3 2 + + n 2 để tính nó thì có nhiều cách The sum of the first n n even integers is 2 2 times the sum of the first n n integers, so putting this all together gives. For n ≥ 0 n ≥ 0, let S(n) S ( n) denote the statement.Technically, it should be called the principal square root of 2, to distinguish it from the negative number with the same property. Share Cite answered Oct 18, 2014 at 15:07 Brad 100 1 9 where did the (−1)k ( − 1) k go between lines 1 and 2 Sep 15, 2022 at 11:33 Add a comment Explanation: using the method of proof by induction. Ex 4. Method 1: You can take a graphical approach to this problem: It can be seen that the graphs meet at (0, 1), 2x 2 x is greater until they intersect when x ≈ 3. The brute force approach: We have. Related. . If we consider n consecutive natural numbers, then finding the sum of the squares of these numbers is represented as Σ i = 1 n i 2. NCERT Solutions for Class 10 Science Chapter 1; NCERT Solutions for Class 10 Science Chapter 2; NCERT Solutions for Class 10 Science Chapter 3 Hint $ $ First trivially inductively prove the Fundamental Theorem of Difference Calculus $$\rm\ F(n) = \sum_{k\, =\, 1}^n f(k)\, \iff\, F(n) - F(n\!-\!1)\, =\, f(n Sum: 2. Steps {3}{2^n} Show More; Description. Of course, one reason for creating the digamma function is to make formulae like this true. View Solution. It contains well written, well thought and well explained computer science and programming articles, quizzes and practice/competitive programming/company interview Questions. So, the answer to your questions are yes and no. 另外一个很好玩的做法. 22n+1−n2 2 2 n + 1 - n 2. 1 2 + 3 2 + 5 2 + Sum of the series 2^0 + 2^1 + 2^2 +…. 18. 5. and RHS = 1 6 (1 + 1)(2 +1) = 1. .2, was Avatar: Frontiers of Pandora - Title Update 1. + 361 = 1330 What is the value of the sum: #(1^2)+(1^2+2^2)+(1^2+2^2+3^2)+.3..+n² program is the same as above. 1. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.+ (2n + 1)^2 = (n + 1)(2n + 1)(2n Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their … This question already has answers here : Proving 13 +23 + ⋯ +n3 =(n(n+1) 2)2 1 3 + 2 3 + ⋯ + n 3 = ( n ( n + 1) 2) 2 using induction (16 answers) Closed 9 years ago. Our task is to create a program that will find the sum of the series.3 gnidecerp dna 1 gniwollof rebmun larutan eht si tI .Call sum_of_squares function with N as input and store the result in sum_of_squares variable. Simplify (2^(n+1))/((2^n)^(n-1)) Step 1. Input: n = 3. Arithmetic. Input: n = 3. The term before in the sum will be half of 2, so we can also write the entire sum as: Find the sum of the series $$1^2-2^2+3^2-4^2+-(2n)^2$$ I tried rewriting it as $$\sum_{r=1}^{2n}-1^{n+1}(r^2)$$ but it didn't help. A term of the form f(n)g(n) can usually be converted to a L'Hopital's rule form by taking the log of both sides. A sequence is an ordered list of numbers. M = 2 1 + 2 + 3 + + n 1 n + 1 ⇒ G. 84.srotcaf nommoc eht gnillecnac yb noisserpxe eht ecudeR . In the statement of the problem we see $1,2,2,3$ but we don't see the next $4$ numbers, which are the solution. Prove the following by using the principle of mathematical induction for all n ∈ N.1. The equation calculator allows you to take a simple or complex equation and solve by best method possible. Step 2: Click the blue arrow to submit and see the result! The equation solver allows you to enter your problem and solve the equation to see the result. Assume is true for some positive integer , then show the relationship is true for , namely that: First note that: which can be written: . Step 3: Calculate the sum of the first n natural number. 第一行1个圈，圈内的数字为1. b) Add the answer from the previous step 6 to the numerator 2. Summing integers up to n is called "triangulation". This is because each successive summand is linear, which makes the growth rate of an a n faster than that and in particular becomes a quadratic.1.. Solve your math problems using our free math solver with step-by-step solutions. 第二行2个圈，圈内的数字都为2，..org or mail your article to review-team@geeksforgeeks. Hint only: For n ≥ 3 you have n2 > 2n + 1 (this should not be hard to see) so if n2 < 2n then consider 2n + 1 = 2 ⋅ 2n > 2n2 > n2 + 2n + 1 = (n + 1)2. = - 1 n - 1 n - 1 + 1 2 + n 2 = - n - 1 n 2 + n 2 = - n 2 - n 2 + n 2 = - n 2 + n + 2 n 2 2 = n 2 + n 2. Step 2. Multiply the exponents in . Step 2. Suppose we take 2^n in the sum. Visit Stack Exchange This question already has answers here : Proving 13 +23 + ⋯ +n3 =(n(n+1) 2)2 1 3 + 2 3 + ⋯ + n 3 = ( n ( n + 1) 2) 2 using induction (16 answers) Closed 9 years ago. . Matrix. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo. 211k 17 17 gold badges 135 135 silver badges 287 287 bronze badges $\endgroup$ 4 $\begingroup$ Wow thanks for this detailed solution! 1/2+2/3 Final result : 7 — = 1. See your article appearing on the GeeksforGeeks main page and help other Geeks. this involves the following steps. Study Materials. S: 1 3 = 1 R. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries.,2,1= k gnittuP 1+k3− 2k3= 3)1−k(− 3k ytitnedi eht redisnoC 2n+ ⋯+ 22+ 21= nS teL rppoT yb deifireV noituloS :)1+k( P ,yb nevig si )1+k( P 6 )1+k2()1+k(k = 2k+. We know since these are powers of two, that the previous term will be half of 2^n, and the term before that a quarter of 2^n. It is the smallest and only even prime number.1.3. )) = 2 /(( + 1. 2.