Question

In a geometric sequence, the common ratio is -5. The sum of the first 3 terms is 147. What is the value of the first term of the sequence?

Answer 1

$\bf \qquad \qquad \textit{sum of a finite geometric sequence} \\\\ \displaystyle S_n=\sum\limits_{i=1}^{n}\ a_1\cdot r^{i-1}\implies S_n=a_1\left( \cfrac{1-r^n}{1-r} \right)\quad \begin{cases} n=\textit{last term's}\\ \qquad position\\ a_1=\textit{first term}\\ r=\textit{common ratio} \end{cases} \\\\[-0.35em] ~\dotfill$

$\bf \begin{cases} r=-5\\ n=3\\ S_3=147 \end{cases} \implies 147=a_1\left( \cfrac{1-(-5)^3}{1-(-5)} \right)\implies 147=a_1\left( \cfrac{1-(-125)}{1+5} \right) \\\\\\ 147=a_1\cdot \cfrac{126}{6}\implies 147=21a_1\implies \cfrac{147}{21}=a_1\implies 7=a_1$