Question

Some Definitions and Other Useful Information For n.EN define the n-th harmonic number, Hn by: Tl 7 1. . Let n E NU (0); n! = (1)(2) (n-1) (n). By convention, 0! Exercises Use induction to prove the following statements. 1. For every n E N, Σ21 k3-2(n+1)2 2. For every integer n2 2, Ση_2 kE1 1-1!

Hi working on number two I need some help pls explain how you solved the problem thanks

Answer 1

For n = 1 L.H.S. = (1)^3 = 1 R.H.S. = (1)^2 (2)^2/4 = 1 times 4/4 = 1 So, n = 1, Statement is true. for n = k, we assume that, the given statement is true. Now, for n = k + 1, 1^3 + 2^3 + 3^3 + .... + k^3 + (k + 1)^3 = k^2(k + 1)^2/4 + (k + 1)^3 = (k + 1)^2 [k^2/4 + (k + 1)] = (k + 1)^2/4 [k^2 + 4k + 4] = (k + 1)^2/4 (k + 2)^2 So, n = k + 1, statement is true. Hence, for all n elementof N, given statement is true. \r\n For n = 1 L.H.S. = (1)^3 = 1 R.H.S. = (1)^2 (2)^2/4 = 4 times 4/4 = 1 So, n = 1, Statement is true. for n = k, we assume that, the given statement is true. Now, for n = k + 1, 1^3 + 2^3
pardon me for first image.