Prove that: `A_n=3^(n+3)-4^(4n+2)` is divided by `11`

 Prove that: `A_n=3^(n+3)-4^(4n+2)` is divided by `11`

Solution

We will prove by Mathematics Induction:

If `n=0` then `A_n=3^3-4^2=27-16=11` It is true that `A_n` is divided with `11`

Assuming that: `A_n` is divided with `11` for `n=k`        `(1)`

We will prove it is true for `n=k+1` then `A_(k+1)` is divided with `11`

From `A_n=3^(n+3)-4^(4n+2)`

            `A_(k+1)=3^(k+4)-4^(4k+6)`

                         `=3.3^(k+3)-4^4.4^(4k+2)`

                        `=3.3^(k+3)-256.4^(4k+2)`

                        `=3.3^(k+3)-3.4^(4k+2)-253.4^(4k+2)`

                        `=3A_k-253.4^(4k+2)`

We knew that: `A_k` is divided with `11`  from equation `(1)`

and, `253` is divided with `11` too.

So, `A_(k+1)` is divided with `11`

Hence,  `A_(k+1)` is divided with `11`

Solution by: Thin Sokkean