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Question
Find the sum of the following geometric series:
\[\frac{2}{9} - \frac{1}{3} + \frac{1}{2} - \frac{3}{4} + . . . \text { to 5 terms };\]
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Solution
Here, a = \[\frac{2}{9} \text { and }r = - \frac{3}{2}\] .
\[S_5 = a\left( \frac{r^5 - 1}{r - 1} \right)\]
\[ = \frac{2}{9}\left( \frac{\left( \frac{- 3}{2} \right)^5 - 1}{\frac{- 3}{2} - 1} \right)\]
\[ = \frac{2}{9}\left( \frac{\left( - \frac{243}{32} \right) - 1}{\frac{- 3}{2} - 1} \right)\]
\[ = \frac{2}{9}\left( \frac{\frac{- 275}{32}}{\frac{- 5}{2}} \right)\]
\[ = \frac{1100}{1440}\]
\[ = \frac{55}{72}\]
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