Question

3 For each positive integer n, define E(n) 2+4++2n (a) Give a recursive definition for E(n). (b) Let P(n) be the statement E(n) nn1)." Complete the steps below to give a proof by induction that P(n) holds for every neZ+ i. Verify P(1) is true. (This is the base step.) ii. Let k be some positive integer. We assume P(k) is true. What exactly are we assuming is true? (This is the inductive hypothesis.) iii. What is the statement P(k 1)? iv. Show that since P(k) is true, so is P(k +1). (This is the inductive step.) (c) We proved in class that 1 +2+-n(n+1)/2. Explain how we can use this fact to give an alternate proof that E(n) n(n+1) for all nez+

Answer 1

E(1) = 2 E(n) = E(n - 1) + 2n forall n greaterthanorequalto 2 (n) (i) E(n) = 2 = 1(1 + 1) therefore p(1) is true (ii) assume that E9k) = k(k + 1) i.e 2 + 4 + ....+ 2k = k (k + 1) (iii) p(k + 1) is the statement that E(k = 1) = (k + 1) (k + 2) (iv) consider E(k + 1) = 2 + 4 = ..... + 2 (k + 1) = 2 + 4 + .....+ 2k + 2k + 2 = k (k + 1) + 2k + 2 = k^2 + 3k + 2 = (k + 1) (k + 2) therefore p(k + 1) is true (c) from lass 1 + 2 + .... + n = n (n + 1)/2 given E(n) = 2 + 4 + ....+_2n = 2 (1 + 2 = ....+ n) = 2 (n (n + 1)/2) E(n) = n (n + 1)