- #1

teroenza

- 195

- 5

## Homework Statement

Prove by Induction the 11^(n+1)+12^(2n-1) is visible by 133.

For all n>1 or n=1

## Homework Equations

## The Attempt at a Solution

I have shown that the base case of n=1 holds.

Then I assumed that the original statement "11^(n+1)+12^(2n-1)is divisible by 133" holds.

Now I need to show the statement holds for n=n+1 ---> 11^(n+2)+12^(2n+1) must be shown to be divisible by 133. I see it equals

11^(n+1)*11+12^(2n-1)*12^2

I pull out 11*12^2 to create something times the original statement (which is divisible)

11*12^2*[11^(n+1)+12^(2n-1)]

multiplied out this equals 12^2*11^(n+2) + 11*12^(2n+1)

I then subtract from this the necessary multiples of 11^(n+2) and 12^(2n+1) to recreate(re-balance) the (n+1) form 6 lines above.

the two terms I subtract are -143*11^(n+2) - 10*12^(2n+1).

If I can show these are divisible by 133, I have completed the proof. I have checked via computer that their sum is divisible for n=1 through 15, but can't seem to find a way to show it.