Question

The sum of an infinite GP with positive terms is 45 and the sum of its first two terms is 25 . Then the second term of the GP is___

Answer 1

Let ' $a$ ' be the first term and ' $r$ ' be the common ration of the GP.
Then, $\frac{a}{1-r}=45$
$\Rightarrow a=45(1-r)$...(i)
Also, $a+a r=25$
$\Rightarrow a(1+r)=25$
$45(1-r)(1-r)=25 \{\because$ from equation $(\mathrm{i})\}$
$1-r^2=\frac{25}{45}=\frac{5}{9}$
$\Rightarrow r^2=\frac{4}{9} \Rightarrow r= \pm \frac{2}{3}$
When $r=\frac{2}{3}$ then $a=45 \times \frac{2}{3}=30$
Second term $=a r=30 \times \frac{2}{3}=20$
where $r=\frac{-2}{3}$, the term of the GP will become negative
Hence, the second term is 20 .