Question

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.

Answer 1

Here, common ratio, r = 3
n^th term, a_n = 486
S_n = 728

\[a_n = 486 \]

\[ \Rightarrow a r^{n - 1} = 486\]

\[ \Rightarrow a \left( 3 \right)^{n - 1} = 486 \]

\[ \Rightarrow a \left( 3 \right)^n = 486 \times 3 \]

\[ \Rightarrow a \left( 3 \right)^n = 1458 . . . \left( i \right)\]

\[\text { Now, } S_n = 728\]

\[ \Rightarrow 728 = a \left( \frac{3^n - 1}{3 - 1} \right) \]

\[ \Rightarrow 728 = \left\{ \frac{a \left( 3 \right)^n - a}{2} \right\}\]

\[ \Rightarrow 1456 = a \left( 3 \right)^{n - 1} - a \]

\[ \Rightarrow 1456 = 1458 - a \left[ \text { From } \left( i \right) \right]\]

\[ \Rightarrow a = 1458 - 1456 \]

\[ \Rightarrow a = 2\]