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 $$ 2^{1/10}+2^{2/10}+\cdots+2^{n/10}=\frac{2^{(n+1)/10}-2^{1/10}}{2^{1/10}-1}. $$

Answer 2

Just rewrite as $\sum_{k=1}^{n} (2^{1/10})^k $ and use the usual geometric series formula.

Answer 3

Let $2^{1/10} = x$ then you are summing $$ x + x^2 + x^3 + \ldots + x^n = x \frac{x^n-1}{x-1} $$