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 + middot middot middot + 1/n(n+1) = 1 - 1/n+1. 1 + 2 + 3 + 4 + middot middot middot + n lessthanorequalto n^2. Prove that n^2 - 1 is divisible by 8 whenever n is an odd positive integer.

Solution

3. When n= 1 since 1 is odd.

then n² -1 = 1² - 1 =0 which is divisible by 8 .

So the result is true for n=1

Let the result be true for n=k where k is an odd positive number and k = 2m+1(say)

So 8| n² -1

=> 8| (2m+1)² -1 as n =k=2m+1

=> 8| 4m² +4m +1-1

=> 8| 4m² + 4m -----(1)

Now we need to prove the result is true for n= k+1= 2(m+1)+1 =2m+3.

So (2m+3)² -1

= 4m² + 12m+ 9 -1

= 4m² + 4m + 8m + 8 which is divisible by 8 as from (1) , it is clear 8|4m² + 4m and clearly 8| 8m+8 as 8m+8 = 8(m+1) and so 8 must divide their sum.

So the result is true for n=k+1 whenever it is assumed to be true for n=k. Also the result is true for n=1.

Hence by the principle of mathematical induction, the result is true for any positive integer n