Question

Evaluate the summation of 2 times negative 2 to the n minus 1 power, from n equals 1 to 7..

Answer 1

Answer: 86

Explanation:

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Answer 2

`\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} \end{cases} \\\\[-0.35em] \rule{34em}{0.25pt}`

`\bf \displaystyle\sum \limits_(n=1)^(7)\stackrel{\stackrel{a_1}{\downarrow }}{2}(\stackrel{\stackrel{r}{\downarrow }}{-2})^(n-1)\implies S_7=2\left( \cfrac{1-(-2)^7}{1-(-2)} \right)\implies S_7=2\left( \cfrac{1-(-128)}{1+2} \right) \\\\\\ S_7=2\left( \cfrac{1+128}{1+2} \right)\implies S_7=2\left( \cfrac{129}{3} \right)\implies S_7=2(43)\implies S_7=86`