Question

Find S7 for the geometric series 2 + -6 + 18 + -54 +…

Answer 1

Notice that the common ratio between each term is -3.

So you have

$S_7=2+(-6)+18+(-54)+\cdots$
$S_7=2((-3)^0+(-3)^1+(-3)^2+(-3)^3+(-3)^4+(-3)^5+(-3)^6)$
$(-3)S_7=2((-3)^1+(-3)^2+(-3)^3+(-3)^4+(-3)^5+(-3)^6+(-3)^7)$

$\implies(1-(-3))S_7=2((-3)^0-(-3)^7)$
$\implies 4S_7=2(1-(-3)^7)$
$\implies S_7=\dfrac{1-(-3)^7}2=1094$