Using mathematical induction prove that $ 11 \cdot 3^n + 3 \cdot 7^n - 6$ is divisible by 8

Question

Prove that $ \phi(n) =11 \cdot 3^n + 3 \cdot 7^n - 6 $ is divisible by 8 for all $n \in N$.

Base: $ n = 0 $

$ 8 | 11 + 3 - 6 $ is obvious.

Now let $\phi(n)$ be true we now prove that is also true for $ \phi(n+1)$.

So we get $ 11 \cdot 3^{n+1} + 3 \cdot 7^{n+1} - 6$ and I am stuck here, just can't find the way to rewrite this expression so that I can use inductive hypothesis or to get that one part of this sum is divisible by 8 and just prove by one more induction that the other part is divisible by 8.

For instance, in the last problem I had to prove that some expression a + b + c is divisible by 9. In inductive step b was divisible by 9 only thing I had to do is show that a + c is divisible by 9 and I did that with another induction, and I don't see if I can do the same thin here.

Answer 1

Suppose $11*3^n + 3*7^n - 6 = 8k$

The $11*3^{n+1} + 3*7^{n+1} - 6 = 11*3^n*3 + 3*7^n*7 - 6$

$=3(11*3^n + 3*7^n-2) + 4*3*7^n $

$= 3(11*3^n + 3*7^n - 6) + 4*3*7^n + 12$

$= 3(8k) + 4(3*7^n + 3)$; $3*7^n$ is odd and $3$ is odd so $(3*7^n + 3)$ is even.

$= 3(8k) + 8(\frac{3*7^n + 3}2) = 8(3k + \frac{3*7^n + 3}2)$.

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Actually I like and am inspired by Bill Dubuques answer.

We want to prove $\phi(n) = 11*3^n + 3*7^n - 6 \equiv 0 \mod 8$

And we know $\phi(n) = 11*3^n + 3*7^n - 6 \equiv 3*3^n + 3*(-1)^n -6 = 3^{n+1} + 3*(-1)^n - 6 \mod 8$.

So it's a matter of showing $f(n) = 3^{n+1} + 3(-1)^n \equiv 6 \mod 8$.

And if we notice $f(n+2) = 3^{n+3} + 3(-1)^{n+2} = 3^{n+1}*9 + 3(-1)^{n} \equiv 3^n + 3(-1)^{n}= f(n) \mod 8$.

So it's now just a matter of showing for $f(0) \equiv f(1) \equiv 6 \mod 8$.

Which is easily verified $3^1 + 3*(-1)^0 =3+3= 6$ and $3^2 + 3*(-1)^1 = 9 -3 = 6$

Answer 2

Setup the same as your current work:

$\dots$

$\dots = 11\cdot 3^{n+1}+3\cdot 7^{n+1}-6 = 11\cdot 3\cdot 3^{n}+ 3\cdot 7\cdot 7^n - 6$

$=33\cdot 3^n + 21\cdot 7^n - 6 = (11+22)\cdot 3^n + (3 + 18)\cdot 7^n - 6$

$=\underbrace{11\cdot 3^n + 3\cdot 7^n - 6}_{\text{should be familiar}} + \underbrace{22\cdot 3^n + 18\cdot 7^n}_{\text{unknown}}$

Now, what can we say about $22\cdot 3^n+18\cdot 7^n$? Anything? You say in a previous example, you had to run a second induction proof to finish, might that be useful here?

Answer 3

$\!\!\bmod 8\!: f(n\!+\!2)\equiv f(n)\,$ by $\,a\equiv 3,7\Rightarrow a^{\large 2}\!\equiv 1\Rightarrow a^{\large n+2}\!\equiv a^{\large n}.\,$ So $\,8\mid f(n)\!\!\iff\!\! 8\mid f(n\!+\!2)$ thus by (strong/parity) induction, it is true for all $n$ $\iff$ it is true for the base cases $\,n=0,1.$