Problem 5. Guess a formula for 1^2 + 3^2 + 5^2 + ... + (2n -...

Problem 5. Guess a formula for $1^2 + 3^2 + 5^2 + ... + (2n - 1)^2$ Hint: • Expand $\sum_{i=1}^{n} (2i - 1)^2$ • use results of Problem 2(Sum of squares) • Gaussian summation • rearrange the results. • Then prove by Induction.

Transcript

00:01 In this problem we are first asked to provide a formula for the sum of the squares of the first n natural numbers.

00:17 So the formula for the first n, the squares of the first n natural numbers is given by 1 squared plus 2 squared plus so on plus n squared which equals to n times n plus 1 times 2 times 2 times 2 times so on plus n squared.

00:34 X x x x the whole divided by 6 we are asked to prove this by induction let us consider this to be represented with p of n so first in induction we need to prove that p of 1 is true so here let us consider n to be 1 in the left hand side so we get 1 squared which is 1 now let us consider n to be 1 in the right hand side so we get 1 squared which is 1 now let us consider n to be 1 in the right hand side we get 1 times 1 plus 1 times 2 times 1 plus 1 the whole divided by 6 simplifying this we have 1 times 1 plus 1 is 2 2 adding with 1 we get 3 the whole divided by 6 1 times 2 is 2 is 2 and 2 times 3 3 is 6 so we get 6 divided by 6 so here we can see that the left hand side and the right hand side are equal to 1.

01:38 So therefore we have proved that p of 1 is true.

01:46 The second part of the induction process involves assuming p of k to be true.

02:00 So therefore let us consider n to be equal to k in the left hand side as well as the right hand side.

02:06 So we get 1 squared plus 2 squared plus so on up to k squared equals to k times k plus 1 times 2 times k plus 1 the whole divided by 6.

02:21 So now let us move to the third part of the induction process which is to prove that p of k plus 1 is true.

02:35 So now that is to show that 1 squared plus 2 squared plus so on up to k plus 1 1 the whole squared equals to k plus 1 multiplied with k plus 1 plus 1 which is k plus 2 times 2 x k plus 1 plus 2 3, sorry plus 1 which is 2 times k plus 3 the whole divided by 6 let us consider the left hand side so the left hand side is 1 squared plus 2 squared plus so on up to k squared plus k squared plus k squared plus the whole squared.

03:19 So from the second part of the induction process we know that 1 squared plus 2 squared plus so on up to k squared equals to k times k plus 1 times 2 k plus 1 the whole divided by 6...

Question

Problem 5. Guess a formula for $1^2 + 3^2 + 5^2 + ... + (2n - 1)^2$ Hint: • Expand $\sum_{i=1}^{n} (2i - 1)^2$ • use results of Problem 2(Sum of squares) • Gaussian summation • rearrange the results. • Then prove by Induction.

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