Hi all.

This is a simple challenge problem I made up myself. Consider the sequence $\displaystyle \left(a_n\right)=(1,2,6,12,36,72,\ldots).$

The first term is 1, and successive terms are formed by multiplying the previous term by 2 and 3 alternately, starting with 2. Thus

$\displaystyle a_1=1$

$\displaystyle a_2=1\cdot2=2$

$\displaystyle a_3=2\cdot3=6$

$\displaystyle a_4=6\cdot2=12$

$\displaystyle a_5=12\cdot3=36$

$\displaystyle a_6=36\cdot2=72$

and so on.

Prove that

$\displaystyle a_n\ =\ \frac{2\sqrt{6^{n-2}}+\sqrt{6^{n-1}}+(-1)^n\left(2\sqrt{6^{n-2}}-\sqrt{6^{n-1}}\right)}2$