Assume n is a positive integer Use induction to prove the fo

Assume n is a positive integer. Use induction to prove the following: 1/1 middot 2+1/2 middot 3 + ... 1/n(n + 1)=1 - 1/n + 1 1 + 2 + 3 + 4 +... lessthanorequalto n^2. Prove that n^2 - 1 is divisible by 8 whenever n is an odd positive integer.

Solution

note : only one question allowed per submission

1. 1/1*2 +1/2*3 + ....... = 1-1/n+1 let p(n) be this statement

for n= 1

1/1*2 = 1-1/2

hene true for n =1

let it be true for n = n

i.e.

1/1*2 +1/2*3 + .......1/(n-1)*n = 1-1/n+1 is true

now to prove it is true for n =n+1

sum of first n terms is

1-1/n+1

now

(1-1/n+1)+ 1/(n+1)*(n+2) =

= n(n+2)+1/(n+1)*(n+2)

=n+1/n+2

= 1- 1/n+2

hene proved true for n+1