Show that 3 · 4^n + 51 is divisible by 3 and 9 for all positive integers n.

Step-by-step explanation:

Hello, please consider the following.

So this is divisible by 3.

Now, to prove that this is divisible by 9 = 3*3 we need to prove that

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 is divisible by 3

and we check what happens for k+1

is divisible by 3 and

is divisible by 3, by induction hypothesis

So, the sum is divisible by 3.

Step 3 - Conclusion

We just prove that is divisible by 3 for all positive integers n.

Thanks

30

step-by-step explanation:

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