# Rewriting sum using summation notation, then finding the sum

I'm supposed to rewrite the sum 1,3,7,15,31...1023 using summation notation, then find the sum. I can tell the value being added begins with 2 then doubles to 4,8,etc. But I'm not sure how to write it in summation notation, I figure it's some sort of polynomial, but I don't know how I would be able to solve for that or anything. Any help would be appreciated.

• If the value being added is supposed to double, then the fifth term should be 31, not 25.
– user694818
Commented Aug 13, 2019 at 22:00
• Sorry, fixed it! Commented Aug 13, 2019 at 22:02

As you can see, and you suggest in your question text, each term is the next power of $$2$$ less $$1$$. Thus, since $$1023 = 2^{10} - 1$$, the sum you want is
\begin{aligned} \sum_{i=1}^{10} (2^i - 1) & = \sum_{i=1}^{10} 2^i - \sum_{i=1}^{10}1 \\ & = (2^{11} - 2) - (10) \\ & = 2046 - 10 \\ & = 2036 \end{aligned}\tag{1}\label{eq1}
Note I got $$\sum_{i=1}^{10} 2^i = 2^{11} - 2$$ using the Geometric series sum of $$a\left(\frac{r^n - 1}{r - 1}\right)$$ where $$a = 2$$ is the first term, $$n = 10$$ is the number of terms, and $$r = 2$$ is the common ratio. Thus, you get $$2(2^{10} - 1) = 2^{11} - 2$$.
• $2^{11}-2=2048-2=204\color{red}6$ Commented Aug 13, 2019 at 22:11