[Math] sum up to nth term with fraction in the power

Question

Is there a formula to express the sum up to the nth term of this:

$$ 2^{(1/10)}+2^{(2/10)}+2^{(3/10)}+…+2^{(n/10)}? $$

I am not a mathematician and use computing algorithm but I am looking for a simplification of my algorithms..

Answer 1

In general, when $a\ne 1$, $$ a^1+a^2+\cdots+a^n=\frac{a^{n+1}-a}{a-1}, $$ and hence $$ 2^{1/10}+2^{2/10}+\cdots+2^{n/10}=\frac{2^{(n+1)/10}-2^{1/10}}{2^{1/10}-1}. $$