Question

Problem #5 [30 Points) Prove each of the following statements by Mathematical Induction. Make sure you check P(1), state P(n+1), and give a complete proof of P(n) = P(n+1), n > 1. A. n.(n+1)(n+2) n 1. P(n): 1. 2+2-3+3.4+...+n·(n + 1) = - to od to be used B. P(n): (1 + a)" > 1+na, a>0,,n>1. ) han de stem

Answer 1

(A) P (n) = 1.2 +2.3 + ... + n(n+1) - n(n+1) (n+2) na! P(1) = 1.2 =2 2 *3 1(1+) (1+2). 2732 Hence, P(1) = 2. Tome let it be true for n. for n=t. P(n+1) = 1.2 +2.3 + ... + n(n+1) + (ntl) (n+2) - n(n+1) (n+2) & (n) (2+2) n(n+1) (n+2) + 3 (n+1) (n+2) 10 - (n+1)(n+2) (n+3) = True for the Hence, true for all nyl.

(B) P(n): (Ita) > it noa azo, n>,l. P(1) - (l+a)' - !+ a = !+ la = 1+ a True for n=l. let it be true for n o t to p(n+1) - (i+a) (Ita) (i+a) > (1+a) (i+na) ntl = (Ithat at naz) = !+ (n+1).a +na? > lt (ntu) a Hence, it is true for ntl. Hence, true for all nar.