Prove that $5^n - 2^n$ is divisible by $3$ for all nonnegative integers $n$ using mathematical induction

Question

Using mathematical induction, prove for all integers n 1 that $5^n - 2^n$ is divisible by 3.

Can someone help me with this?

Answer 1

Hint: Note that $5^{n+1}-2^{n+1} = 5\cdot 5^n - 2\cdot 2^n = 5(5^n-2^n) + 3\cdot2^n$.

Answer 2

This one could be solved by induction but is it much neater if you use that \[ a^n-b^n = (a-b) \cdot \sum_{i=0}^{n-1} a^i\cdot b^{n-1-i} \] If you use this one in your problem you see that \[ 5^n- 2^n = (5-2) \cdot \sum_{i=0}^{n-1} 5^i \cdot 2^{n-1-i}=3 \cdot \sum_{i=0}^{n-1} 5^i \cdot 2^{n-1-i}\] Which can be dived by $3$.

If you really want to use Induction use that \[ 5=3+2\] (ok this hint seems ridicolous it shall be used like this) \[ 5^{n+1} - 2^{n+1} = (3+2) \cdot 5^n - 2 \cdot 2^n= 3 \cdot 5^n+ 2(5^n-2^n)\]