Find the $12^{\text{th}}$ term of a G.P. with common ratio 2, whose $8^{\text{th}}$ term is 192.                                      [Page No. 193]

In a GP, $T_n = ar^{n-1}$

$$\text{Given: } T_8 = ar^7 = 192, r = 2$$

$$\text{So, } a(2^7) = 192$$

$a = \frac{128}{192} = \frac{192}{128} = \frac{3}{2}$

$$\begin{aligned} \text{Now,} \quad T_{12} &= a(2^{11}) \\ &= \frac{3}{2} \times 2048 = 3072 \end{aligned}$$

Therefore, the 12th term is $3072$.

Caution
Students should not confuse an arithmetic progression (AP) with a geometric progression (GP). An AP has a common difference that is added each time, while a GP has a common ratio that is multiplied each time.