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प्रश्न
The common ratio of a G.P. is 3 and the last term is 486. If the sum of these terms be 728, find the first term.
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उत्तर
Here, common ratio, r = 3
nth term, an = 486
Sn = 728
\[a_n = 486 \]
\[ \Rightarrow a r^{n - 1} = 486\]
\[ \Rightarrow a \left( 3 \right)^{n - 1} = 486 \]
\[ \Rightarrow a \left( 3 \right)^n = 486 \times 3 \]
\[ \Rightarrow a \left( 3 \right)^n = 1458 . . . \left( i \right)\]
\[\text { Now, } S_n = 728\]
\[ \Rightarrow 728 = a \left( \frac{3^n - 1}{3 - 1} \right) \]
\[ \Rightarrow 728 = \left\{ \frac{a \left( 3 \right)^n - a}{2} \right\}\]
\[ \Rightarrow 1456 = a \left( 3 \right)^{n - 1} - a \]
\[ \Rightarrow 1456 = 1458 - a \left[ \text { From } \left( i \right) \right]\]
\[ \Rightarrow a = 1458 - 1456 \]
\[ \Rightarrow a = 2\]
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