Question

The fourth term of a G.P. is eight times its seventh term. The fifth term of then G.P. is `3/16`, then find its 12^th term.

Answer 1

By formula,

\[a_n = ar^{n \: - \: 1}\]

Given,

The fourth term of a G.P. is eight times its seventh term.

\[a_4 = 8a_7\]

\[ar^{4 \: - \: 1} = 8ar^{7 \: - \: 1}\]

\[ar^3 = 8ar^6\]

\[\frac{r^6}{r^3} = \frac{a}{8a}\]

\[r^3 = \frac{1}{8}\]

\[r = \sqrt[3]{\frac{1}{8}} = \frac{1}{2}\]

Given,

Fifth term \[{} = \frac{3}{16}\]

⇒ \[ar^4 = \frac{3}{16}\]

⇒ \[a \times \left(\frac{1}{2}\right)^4 = \frac{3}{16}\]

⇒ \[a \times \frac{1}{16} = \frac{3}{16}\]

⇒ \[a = 3\]

12th term:

\[a_{12} = ar^{12 \: - \: 1}\]

\[{} = ar^{11}\]

\[{} = 3 \times \left(\frac{1}{2}\right)^{11}\]

Hence, \[12^{\text{th}}\] term \[{} = 3 \times \left(\frac{1}{2}\right)^{11}\]