Find S, for the geometric sequences: 4,-20, 100

Danelle Albright 2022-01-07 Answered
Find S, for the geometric sequences: \(\displaystyle{4},-{20},{100}\)

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kalfswors0m
Answered 2022-01-08 Author has 960 answers
The common ration of the sequence is
\(\displaystyle{r}={\frac{{−{20}}}{{{4}}}}=−{5}{r}\)
\(\displaystyle={\frac{{{100}}}{{−{20}}}}=−{5}\)
Since, the value of the \(\displaystyle{r}{ < }{1}\), the formula for the sum of the n term of the series is:
\(\displaystyle{S}_{{n}}={\frac{{{a}{\left({1}−{r}{n}\right)}}}{{{1}−{r}}}}\)
Then, sum of first five terms is
\(\displaystyle{S}_{{5}}={\frac{{{4}{\left({1}−{\left(−{5}\right)}{\left\lbrace{5}\right\rbrace}\right)}}}{{{1}−{\left(−{5}\right)}}}}\)
\(\displaystyle={\frac{{{4}{\left({1}+{3125}\right)}}}{{{1}+{5}}}}\)
\(\displaystyle={4}{\frac{{{3126}}}{{{6}}}}\)
\(\displaystyle={4}{\left({521}\right)}={2084}\)
Thus, the sum of the first five terms is 2084.
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