Question

If the sum of first two terms of an infinite GP is 1 every term is twice the sum of all the successive terms, then its first term is
(a) 1/3
(b) 2/3
(c) 1/4
(d) 3/4

Answer 1

(d) 3/4

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{terms}\mathrm{of}\mathrm{the}\mathrm{G}.\mathrm{P}.\mathrm{be}a,a_2,a_3,a_4,a_5,...,\infty . \\ \mathrm{And},\mathrm{let}\mathrm{the}\mathrm{common}\mathrm{ratio}\mathrm{be}r. \\ \mathrm{Now},a+a_2=1 \\ \therefore a+ar=1........\left(\mathrm{i}\right) \\ \mathrm{Also},a=2\left(a_2+a_3+a_4+a_5+...\infty \right) \\ \Rightarrow a=2\left(ar+a{r}^2+a{r}^3+a{r}^4+...\infty \right) \\ \Rightarrow a=2\left(\frac{ar}{1-r}\right) \\ \Rightarrow 1-r=2r \\ \Rightarrow 3r=1 \\ \Rightarrow r=\frac{1}{3} \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{r}\mathrm{in}\left(\mathrm{i}\right): \\ a+\frac{a}{3}=1 \\ \Rightarrow \frac{4a}{3}=1 \\ \Rightarrow 4a=3 \\ \Rightarrow a=\frac{3}{4} \end{gathered}$$