Answer:
Step-by-step explanation:
Hello, please consider the following.
[tex]3\cdot 4^n+51=3\cdot 4^n+3\cdot 17=3(4^n+17)[/tex]
So this is divisible by 3.
Now, to prove that this is divisible by 9 = 3*3 we need to prove that
[tex]4^n+17[/tex] is divisible by 3. We will prove it by induction.
Step 1 - for n = 1
4+17=21= 3*7 this is true
Step 2 - we assume this is true for k so [tex]4^k+17[/tex] is divisible by 3
and we check what happens for k+1
[tex]4^{k+1}+17=4\cdot 4^k+17=3\cdot 4^k + 4^k+17[/tex]
[tex]3\cdot 4^k[/tex] is divisible by 3 and
[tex]4^k+17[/tex] is divisible by 3, by induction hypothesis
So, the sum is divisible by 3.
Step 3 - Conclusion
We just prove that [tex]4^n+17[/tex] is divisible by 3 for all positive integers n.
Thanks
