Question

Use the formula to evaluate the series -3+6-12+24-...-a7

Answer 1

-3+6-12+24   <---- notice the terms firstly, they go as

-3 , +6 , -12 , +24 ,....    <---- to get the next term's value, you multiply it by -2

thus, is a geometric sequence, and -2 is the "common difference", and the first term is -3 of course.

so... what's the sum of the first 7 terms?


$\bf \qquad \qquad \textit{sum of a finite geometric sequence}\\\\ S_n=\sum\limits_{i=1}^{n}\ a_1\cdot r^{i-1}\implies S_n=a_1\left( \cfrac{1-r^n}{1-r} \right)\quad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ r=\textit{common ratio}\\ ----------\\ a_1=-3\\ r=-2\\ n=7 \end{cases}$

$\bf S_7=\sum\limits_{i=1}^{7}\ -3\cdot (-2)^{i-1}\implies S_7=-3\left( \cfrac{1-(-2)^7}{1-(-2)} \right) \\\\\\ S_7=-3\left( \cfrac{1-(-128)}{1+2}\right)\implies S_7=-3\left( \cfrac{129}{3} \right)\implies S_7=-3(43) \\\\\\ S_7=-129$