7^n - 3^n is divisible by 4

Question

Need a worked solution please, it's mathematical induction

Answer 1

Step 1: Test n=1, we get 7-3=4=4x1

Step 2: Assume n=k, 7^k-3^k is divisible by 4 which means 7^k-3^k=4N, where N is a positive integer

Step 3: Prove for n=k+1; 7^k+1-3^k+1=4Q, where Q is a positive integer

 

7*7^k-3*3^k+1, Now break 7*7^k into 4*7^k+3*7^k-3*3^k, Factor out 3 from the last two terms, then we get the following;

 

4*7^k+3(7^k-3^k). Remember from our earlier assumption we assumed that 7^k-3^k was divisible by 4 so, we get 

 

4*7^k+3*4N The next step is factor the 4, then we get 4(7^k+3N)=4Q, where Q=7^k+3N

 

Proven by induction.