Question

Write the series using sigma notation with lower limit n=4.

Answer 1

`\begin{array}{cccccccccc} &(-1)/(4)\left( -(1)/(2) \right)&(1)/(8)\left( -(1)/(2) \right)&(-1)/(16)\left( -(1)/(2) \right)&(1)/(32)\left( -(1)/(2) \right)\\ -\cfrac{1}{4}&\cfrac{1}{8}&-\cfrac{1}{16}&\cfrac{1}{32}&-\cfrac{1}{64}... \end{array}\hspace{5em}\stackrel{\textit{common ratio}}{r=-(1)/(2)} \\\\[-0.35em] ~\dotfill`

`\qquad \qquad \textit{sum of a finite geometric sequence} \\\\ \displaystyle S_n=\sum\limits_(i=1)^(n)\ a_1\cdot r^(i-1)\qquad \begin{cases} n=\textit{last term's}\\ \qquad position\\ a_1=\textit{first term}\\ r=\textit{common ratio}\\[-0.5em] \hrulefill\\ r=-(1)/(2)\\ a_1=-(1)/(4) \end{cases}\implies \sum_(n=1)^(n=4)~-(1)/(4)\left( -(1)/(2) \right)^(n-1)`