Question

How to Solve? I am completely stuck on this assignment and I haven't found an answer yet.

∑10 n=1 4(1/4)^n-1

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=10\\ a_1=4\\ r=(1)/(4) \end{cases}`

`{\displaystyle\sum_(n=1)^(10)}~4\left( (1)/(4) \right)^(n-1)\implies 4\left( \cfrac{1-\left( (1)/(4) \right)^(10)}{1-(1)/(4)} \right)\implies 4\left( \cfrac{(1048575)/(1048576)}{(3)/(4)} \right) \\\\\\ 4\left( \cfrac{349525}{262144} \right) \implies \cfrac{349525}{65536} ~~ \approx ~~ 5.33`