Question

Topic: Finite Geometric Sums what is the value of ..

Answer 1

$\bf \sum\limits_{i=a_1}^{a_n}\ a1\cdot (r)^{n-1}\iff S_n=a_1\left( \cfrac{1-r^n}{1-r} \right)\qquad \begin{cases} a_1=\textit{first term}\\ r=\textit{common ratio}\\ n=n^{th}\ term \end{cases}\\\\ -----------------------------\\\\ \begin{array}{llll} \sum\limits_{i=1}^{3}\ &4\cdot \left( \frac{1}{2} \right)^{i-1}\\ &\uparrow \quad \uparrow \\ &a_1\quad r \end{array}\iff S_3=4\left( \cfrac{1-\left( \frac{1}{2} \right)^3}{1-\left( \frac{1}{2} \right)} \right)$