Question

In a G.P.,the product of the first four terms is 4 and the second term is reciprocal of the fourth term. The sum of infinite terms of the G.P. is

• A. -8
• B.
• C.
• D. 8

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A

-8

B

C

D

8

To find the sum of the series we have to find the common ratio r and first term a.

When we apply the given conditions, r and a may be complex also.

We will use these complex values of a and r,

because there is no condition given in the question like the G.P. is real.

Let the four terms be $\frac{a}{r^3}$, $\frac{a}{r}$, ar, a$r^3$

Product = $\frac{a}{r^3}$ $\times$ $\frac{a}{r}$ $\times$ ar $\times$ a$r^3$

4 = $a^4$

$\Rightarrow$ $a^2$ = $\mp$ 2 (a can be a complex number)

$t_2$ = $\frac{1}{t_4}$ (given)

$\Rightarrow$ $\frac{a}{r}$ = $\frac{1}{a{r}^3}$

$\Rightarrow$ $a^2$ = $\frac{1}{r^2}$

$r^2$ = $\frac{1}{a^2}$

= $\frac{\mp 1}{2}$ (r can be a complex number)

Sum to infinity

S = $\frac{a}{r^3(1-r^2))}$ [common ratio = $r^2$ first term = $\frac{a}{r^3}$

$a=\frac{\mp 1}{r}$

$\Rightarrow$ S = $\frac{\mp 1}{r\times r^3(1-r^2)}$

= $\frac{\mp 1}{r^4(1-r^2)}$

Taking $r^2$ = $\frac{1}{2}$,

S= $\frac{\mp 1\times 2^2}{1-\frac{1}{2}}$

$=\mp 8$

Taking $r^2$ = $\frac{-1}{2}$,

S = $\frac{\mp 1\times 2^2}{1+\frac{1}{2}}$

= $\frac{\mp 8}{3}$

$\Rightarrow$ S can be 8 or -8 or $\frac{-8}{3}$ or $\frac{8}{3}$