Question

The sum of first three terms of a G.P. is $\frac{13}{12}$ and their product is $-1$. Then, which of the following statements is incorrect?

• A. Common ratio is $\frac{-3}{4}$ or $\frac{-4}{3}$
• B. First three terms are $\frac{4}{3},-1,\frac{3}{4}$ for $r=\frac{-3}{4}$
• C. First three terms are $\frac{3}{4},-1,\frac{4}{3}$ for $r=\frac{-4}{3}$
• D. Common ratio is $\frac{3}{4}$ or $\frac{4}{3}$

Answer 1

Answer: (D)

Let $\frac{a}{r},a,ar$ be the three terms of the G.P.
Then, $\frac{a}{r}+a+ar=\frac{13}{12}.....$(i)
and $\frac{a}{r}\cdot a\cdot ar=-\mathrm{1.....}$(ii)
From Eq. (ii), we get $a^3=-1$
$\Rightarrow a=-1$ (considering only real roots)
Substituting $a=-1$ in Eq. (i), we have
$\frac{-1}{r}-1-r=\frac{13}{12}$
$\Rightarrow r^2+r+1+\frac{13r}{12}=0$
$\Rightarrow 12r^2+25r+12=0$
On solving, we get
$r=\frac{-3}{4}\text{ or }\frac{-4}{3}$
Thus, the three terms of G.P. are $\frac{4}{3},-1,\frac{3}{4}$
for $r=\frac{-3}{4}$ and $\frac{3}{4},-1,\frac{4}{3}$ for $r=\frac{-4}{3}$.