Question

Write the statements for the basis, the induction hypothesis, and the induction step for the questions below.

1. 1 + 2 +…+n = n(n+1)/2


2. 1/1*2 + 1/2*3+…+1/n*(n+1) = n/n+1

Answer 1

for k=1, 1 = 1(2)/2
hyp: as shown
1+...+ k + (k+1) = k(k+1)/2 + (k+1)
= [k(k+1) + 2(k+1)]/2
= (k+1)(k+2)/2

for k=1, 1/1*2 = 1/2

1/1*2 + ... + 1/k(k+1) + 1/(k+1)(k+2)
= k/(k+1) + 1/(k+1)(k+2)
= (k(k+2) + 1)/(k+1)(k+2)
= (k+1)^2 / (k+1)(k+2)
= (k+1)/(k+2)