Question

A Geometric series has a first term of 4 and a common ratio of 6. what formula is equivalent to the partial sum of the first n terms of the series.

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}\\ ----------\\ a_1=4\\ r=6 \end{cases} \\\\\\ S_n=4\left( \cfrac{1-6^n}{1-6} \right)\implies S_n=4\left( \cfrac{1-6^n}{-5} \right)`