Originally Posted by

**princemac** Hi everyone I need a little help, Im stuck.

I have to prove by induction that:

$\displaystyle

2^{n+2} + 3^{2n+1}

$

is exactly divisible by 7 for all positive integers of n.

Now what I have done is:

1)

If $\displaystyle 2^{n+2} + 3^{2n+1}$ is divisible by 7

=>( $\displaystyle 2^{n+2} + 3^{2n+1}=7a$ )-----> Pn statement

where a is a positive integer

2) $\displaystyle P_1$ statement where n=1

=>

LHS: $\displaystyle 2^{1+2} + 3^{2(1)+1} = 35$

RHS: $\displaystyle 7(5) = 35$

Therefore $\displaystyle P_1$ is true.

3) Assuming $\displaystyle P_k$ to be true where n=k

=> $\displaystyle P_k = 2^{k+2} + 3^{2k+1} = 7a$

4) Therefore

$\displaystyle P_{k+1} = 2^{(K+1)+2} + 3^{2(k+1)+1}$

Now here is where Im stuck, Im not getting to factor out a 7 i.e.

getting 7(some integer) to prove that $\displaystyle P_{k+1}$ is divisible by 7.

Any help would be greately appreciated.