Prove the following by induction For all n E N 7n 2n is divi

Prove the following by induction.

For all n E N, 7^n 2^n is divisible by 5.

In mathematical induction we follow below steps to prove any given expression.

1.check if the expression satisfies the condition n=1, if yes go for step 2.

2.check if it satisfies the condition for n=n or for some n=k, if yes go for step 3.

3.finally check if it satisfies n=n+1.

Solution: Here we if 7^n-2^n is divisible by 5 then the result should be a multiple of 5

so assuming 7^n-2^n=5m---> equation 1

1.For n=1, we get 7-2=5 , which is multiple of 5 and divisible by 5. so moving towards condition 2.

2.if 7^n-2^n is divisible by 5 then for some k E N, 7^k-2^k =5m, which is also a multiple of 5 and so hence condition 2 also gets satisfied.

3.taking n=n+1, substitute n=n+1 in the expression , 7^n+1-2^n+1

as we have 7^n =5m+2^n from equation1, substituting it in above , we get

7^n+1-2^n+1

=7^n.7-2^n.2

=7(5m+2^n)-2^n.2

=35m+7.2^n-2^n.2

=35m+2^n(7-2)

=35m+5.2^n

=5(7m+2^n)

=5M

now assuming 7m+2^n as some M we have 7^n+1-2^n+1=5M , where 5M is multiple of 5 and hence divisible by 5.

Therefore the given expression satisfies all the conditions of mathematical induction.

Hence proved.