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( (1)/(2) \right)^(i-1)\\ &\uparrow \quad \uparrow \\ &a_1\quad r \end{array}\iff S_3=4\left( \cfrac{1-\left( (1)/(2) \right)^3}{1-\left( (1)/(2) \right)} \right)`