Question

What is the first term in a geometric sequence if the common ratio is − 2 and the sum of the first six terms is −105?

Answer 1

$\bf \qquad \qquad \textit{sum of a finite geometric sequence} \\\\ S_n=\sum\limits_{i=1}^{n}\ a_1\cdot r^{i-1}\implies S_n=a_1\left( \cfrac{1-r^n}{1-r} \right)\quad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ r=\textit{common ratio}\\ ----------\\ r=-2\\ n=6\\ S_6=-105 \end{cases}$

$\bf -105=a_1\left( \cfrac{1-(-2)^6}{1-(-2)} \right)\implies -105=a_1\left( \cfrac{1-(64)}{1+2} \right) \\\\\\ -105=a_1\left( \cfrac{-63}{3} \right)\implies -105=a_1(-21) \\\\\\ \cfrac{-105}{-21}=a_1\implies 5=a_1$