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Inductive, So does it hold true for n=k+1

3 raised 2(k+1)+1 +2 raised(k+1)-1 = 3 raised 2k+2+1 +2 raised (k+1)-1

= 3 raised 2k+1 x 3 raised2 + 2 raised k x 2 raised 0

=9 x 3 raised 2k+1 + 1 x 2 raised k

= 27 x 3 raised 2k +1x2 raised k

Problem isn't posting correctly

- Apr 22, 2018

- 251

$3^{2(k+1)+1}+2^{(k+1)-1}$

$=\quad3^{2k+3}+2^k$

$=\quad9\cdot3^{2k+1}+2\cdot2^{k-1}$

$=\quad7\cdot3^{2k+1}+2\cdot\left(3^{2k+1}+2^{k-1}\right).$

It should be straightforward to proceed from here.

When doing problems of this kind, look at the number you want your expression to be divisible by (in this case $7$) and try and rearrange your expression to involve it.