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Question
The fifth term of a G.P. is 81 whereas its second term is 24. Find the series and sum of its first eight terms.
Solution
Let a be the first term and r be the common ratio of the G.P.
\[a_2 = 24 \]
\[ \Rightarrow a r^{2 - 1} = 24\]
\[ \Rightarrow ar = 24 . . . \left( i \right)\]
\[\text { Similarly }, a_5 = 81 \]
\[ \Rightarrow a r^{5 - 1} = 24\]
\[ \Rightarrow a r^4 = 81\]
\[ \Rightarrow \frac{24 \times r^4}{r} = 81 \left[ \text { From } \left( i \right) \right]\]
\[ \Rightarrow r^3 = \frac{81}{24} \]
\[ \therefore r^3 = \frac{27}{8}\]
\[ \Rightarrow r = \frac{3}{2}\]
\[\text { Putting }r = \frac{3}{2}\text { in } \left( i \right)\]
\[3a = 48 \]
\[ \Rightarrow a = 16\]
\[\text { So, the geometric series is } 16 + 24 + 36 + . . . + 16 \left( \frac{3}{2} \right)^8 \]
\[\text { And }, S_8 = 16\left( \frac{\left( \frac{3}{2} \right)^8 - 1}{\frac{3}{2} - 1} \right) \]
\[ \Rightarrow S_8 = 32\left( \frac{6561 - 256}{256} \right) = \frac{32 \times 6305}{256} = \frac{6305}{8}\]
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