Question

1. Prove the following statement by mathematical induction. For all positive integers n. 2++ n+1) = 2. Prove the following statement by mathematical induction. For all nonnegative integers n, 3 divides 22n-1. 3. Prove the following statement by mathematical induction. For all integers n 27,3" <n!

Answer 1

kokt to, ka 1+1 8.1) For not 12 = (1 - 2 holds for nat Suppose it is holds for n=k, K(K+2+1 (kanen toost I k . K(K+1) k+1 consider for n=k+1 han'+ za tist(X+0 (R+7410) *** t. +- + 12 2.3 RERAD (K+D(K+1+1) = (**)4 (omaxpurus) – By (1) kt1 Kt2 ktl/ KA2 (+)² = kt! (KHU(R+2) (k+1+1 .: it is hold for n=k+1 & positive integersn, I t I trust a n "! 12 2:3 n(n+1) n+1 holds 8.2) For no = 221170 3 divides o For nat =) 22h 1 = 3 . :.3 divides 3 u given statement is true for não & n=1 suppose it is true for n=k ie, 3 divides 224_1 : 22k, = 3K, (Kiez) tid consider for n=k+1 2ktv 2K+2 21 = 4224 = 42 2 + 4 +3 = 4( 22R 1) +3 = 4(36,2+3 - By (ii) = 3(4K,+1) - 1 = 342 – (where K2 = 4K, +1 EZ) 21K+D 2

3 divide Statment hold 22(R+12 for n=k+1 21 . For & non-negative integers n 3 divides 20). 8.3) For n=7 37 - 2187 & 71=5040 37 <7? n=7 statements hold. suppose an For n = 7+k Statement holds je 57+K), (7 +)! — (i) consider, n = 7+(K+1) 7 +K I 3 :3 7317+K?! < (7+*+1X7+)! (377 +*+1) 7+(8+1) < (7 +*+0! (for n= 7+k) ign En! & integers wit, 3rn?