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

Find S, for the geometric sequences: $$\displaystyle{4},-{20},{100}$$

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kalfswors0m
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.