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Question
The nth term of a G.P. is 128 and the sum of its n terms is 225. If its common ratio is 2, then its first term is
Options
(a) 1
(b) 3
(c) 8
(d) none of these
Solution
\[a_n = 128, S_n = 225 \text{ and } r = 2\]
\[ a_n = 128\]
\[ \therefore a r^\left( n - 1 \right) = 128\]
\[ \Rightarrow 2^\left( n - 1 \right) a = 128\]
\[ \Rightarrow \frac{2^n a}{2} = 128\]
\[ \Rightarrow 2^n = \frac{256}{a} . . . . . . . . (i)\]
\[\text{ Also }, S_n = 225\]
\[ \Rightarrow a\left( \frac{r^n - 1}{r - 1} \right) = 225\]
\[ \Rightarrow a\left( \frac{2^n - 1}{2 - 1} \right) = 225\]
\[ \Rightarrow a\left( \frac{256}{a} - 1 \right) = 225 \left[ \text{ Using } (i) \right]\]
\[ \Rightarrow 256 - a = 225\]
\[ \Rightarrow a = 256 - 225\]
\[ \Rightarrow a = 31\]
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