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

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jeew-m | College Teacher | (Level 1) Educator Emeritus

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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.

 

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