<s>[INST] Prove that $5^n - 2^n$ is divisible by $3$ for all nonnegative integers $n$ using mathematical induction Using mathematical induction, prove for all integers n  1 that $5^n - 2^n$ is divisible by 3.
Can someone help me with this?
 [/INST] 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)
\[ 5^{n+1} - 2^{n+1} = (3+2) \cdot 5^n - 2 \cdot 2^n= 3 \cdot 5^n+ 2(5^n-2^n)\]
</s>