Without using a calculator, how do I find:
$$\sum_{k=0}^7 \left(\tfrac{2}{3}\right)^k $$
I know about the formula for the sum of a geometric series $\frac{a\cdot(1-r^n)}{(1-r)}$ but im not sure how to apply it. I feel as if you make a small series and try to formulate a equation out of it, such as:
$$ \left(\tfrac{2^0}{3^0}\right)+\left(\tfrac{2^1}{3^1}\right)...\left(\tfrac{2^7}{3^7}\right)$$
and pull out some value. I'm lost after that. Any help?
$\endgroup$ 11 Answer
$\begingroup$So that formula for the sum of geometric series was
$$\sum_{i=0}^{n-1} a r^i = a \frac{1-r^n}{1-r}$$
Compare this to $$\sum_{i=0}^{7} (-2/3)^i$$ to identify what is a, r and n. Then plug those in.
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