My birthday puzzle (in year 2015)

Frankly, I just typed 38 seeing your age in your profile , and it came out to be correct!! BTW happy birthday. ( sorry I don't know in advance or belated).

$\large{3 \left(\sum \frac{n^3}{2^{n+1}} -\sum \frac{2}{3.2^{n+1}}\right)}$

For the former part , we know

$\large{ \sum _{n=1}^{\infty} x^n= \frac{x}{1-x}}$ with $|x|<1$

Differentiate it thrice and each time after differentiating multiply it by $x$ and then again differentiate. (In total 3 times)

We get $\large{\sum n^3 x^{n-1}=\frac{1+4x+x^2}{(1-x)^4}}$

Now multiply by $x^2$ and put $x=1/2$

The latter part is a simple G.P.

Hence the answer is $\boxed{38}$