The second term of a GP is 24 and its fifth term is 81.
Question:

The second term of a GP is 24 and its fifth term is 81. Find the sum of its first five terms.

Solution:

Given: second term of a GP is 24 and its fifth term is 81.

To find: sum of first five terms of the G.P.

$a r=24 \& a r^{4}=81$

dividing these two terms we get:

$\Rightarrow \frac{\mathrm{ar}^{4}}{\mathrm{ar}}=\frac{81}{24}$

$\Rightarrow \Gamma^{3}=\frac{27}{8}$

Taking cube root on both the sides we get,

$\Rightarrow r=\frac{3}{2}$

Substituting this value of r in ar = 24 we get

$a=24 /(3 / 2)=(24 \times 2) / 3=16$

$\therefore$ Sum of first Five terms of a G.P. $=a\left(r^{n}-1\right) /(r-1)$

$=16 \times \frac{\left(\frac{3}{2}\right)^{5}-1}{\frac{3}{2}-1}=16 \times \frac{\frac{243}{32}-1}{\frac{3}{2}-1}$

$=16 \times \frac{242 \times 2}{32 \times 1}=242$

Ans: 242