Question

Find a1 when S7=5465, r=3, n=7 using geometric series

Answer 1

`\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}\\[-0.5em] \hrulefill\\ n=7\\ r=3 \end{cases} \\\\\\ S_7=a_1\left( \cfrac{1-r^7}{1-r} \right)\implies 5465=a_1\left( \cfrac{1-3^7}{1-3}\right)\implies 5465=a_1(1093) \\\\\\ \cfrac{5465}{1093}=a_1\implies \boxed{5=a_1}`