P(n) = 5^n - 5 is divisible by 4. is

We can use mathematical induction for
this.

P(n) = 5^n - 5

When
n=1;

p(1) = 5^1-5 = 0=0/4 ; this is divisible by
4.

Let us assume a positive integer p
where

p(p) = 5^p-5

We assume
this is divisible by 4.

So we can
write;

p(p)=(5^p-5) =
4k------(1)

where k is a positive
integer.

so p(p) = 4k

Then
consider n=p+1

p(p+1) =
5^(p+1)-5------(2)

We must show equation (2) is
divisible by 4.

Then p(p+1)=4q where q is a positive
integer.

(1)*5

5p(p) =
5(5^p-5)

         =
5^(p+1)-25

         =
[5^(p+1)-5]-20

5p(p) =
[5^(p+1)-5]-20

5*4k  =
p(p+1)-20

p(p+1) = 5*4k-20=4(5k-5) =
4q

So when n=p+1 p(n) is divisible by
4.

So for all positive n p(n) is divisible by
4.