I have a sum of the form c¹+c²+...cⁿ. Is it possible to obtain a closed formula for this, and if so how?
$\endgroup$ 12 Answers
$\begingroup$You're probably looking for Faulhaber's formula.
Revised answer after question edit: What you've written is the sum of a geometric sequence. A closed form of the sum is:$$\sum_{k=1}^n C^k=\frac{C(C^n - 1)}{C-1}$$
$\endgroup$ 2 $\begingroup$Micapps answered the original question $1^n + 2^n + ...$ but then it was edited to $c^1 + c^2 + ...$
The answer to the edited question seems to be:
$$\sum_{p=1}^n c^p=\frac{1 - c^{(n+1)}}{1 - c} - 1$$
Which is based on this formula I got from :
$$\sum_{n=0}^{N-1} r^n=\frac{1 -r^N}{1 - r}$$
Changes:
- The original starts on n=0 instead of 1, so I can subtract that (assuming c^0 is always 1... not imaginary maybe)... now it's a different formula with a different result.
- and I prefer the "n+1" on the right to the "N-1" on the left side, which is just cosmetic.
