Question

8. Use mathematical induction to prove that n + + 7n 15 3 5 is an integer for all integers n > 0.

Answer 1

Step 1 check n=1 let Pens = n3 tus = ņs the + + Tn is an Integer 15 for P(ID = £ t f at 7/15 P(1) is an Integer let P (K) is an Integer 5+3+?=1 15 Step II P (k) = K3 KS 5 + Ik is an Integer t 3 is so in Check p (kt) is an Integer os not (kf 1² +(k+s +7(441) 3

Ple+1) (601) [ 14.4133 + fars + 1)] K3+1 + 3k (14) Scots & Sc, k" + Sc, K² +56₂ k² 45 Cuk! +5 egiten thet 15 IS 3 5 5k4 + lok3 +34 (kt12 + 3 5 + 1062 '+ 5k + 16 1 5 5 s FC + S +2:) 46++) + kk+11+ k" + 2 +3 +24² .tk (k3+k5 7K t 6+ 2K k3 tks t 3 Pekti at k" + 2k3 +342 +2kt 3 5 5. from equation o - Integer Š To and k = Integer so k" + 2 k 34 36²f2k + 1 = Integer so Plkt) is an Integer By Mathematical Induction PER) [P (n) = hans is also an Integer 35 + In is