Question

What is the sum of the first 7 terms of the series?

−3+6−12+24−... ?

Answer 1

`\bf -3~~,~~\stackrel{-3(-2)}{6}~~,~~\stackrel{6(-2)}{-12}~~,~~\stackrel{-12(-2)}{24}~~,~~...\impliedby \stackrel{\textit{common ratio}}{-2=r} \\\\[-0.35em] ~\dotfill\\\\ \qquad \qquad \textit{sum of a finite geometric sequence} \\\\ S_n=\sum\limits_(i=1)^(n)\ a_1\cdot r^(i-1)\implies S_n=a_1\left( \cfrac{1-r^n}{1-r} \right)\quad \begin{cases} n=n^(th)\ term\\ a_1=\textit{first term's value}\\ r=\textit{common ratio}\\[-0.5em] \hrulefill\\ a_1=-3\\ n=7\\ r=-2 \end{cases}`

`\bf S_7=-3\left[\cfrac{1-(-2)^7}{1-(-2)} \right]\implies S_7=-3\left[\cfrac{1-(-128)}{1+2} \right] \\\\\\ S_7=-3\left( \cfrac{1+128}{3} \right)\implies S_7=-3\left( \cfrac{129}{3} \right)\implies S_7=-3(43) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill S_7=-129~\hfill`