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Q.
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?
Sequences and Series
Solution:
Let $\frac{a}{r}, a, a r$ be the three terms of the G.P.
Then, $ \frac{a}{r}+a+a r=\frac{13}{12} .....$(i)
and $ \frac{a}{r} \cdot a \cdot a r=-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
$\frac{-1}{r}-1-r =\frac{13}{12}$
$\Rightarrow r^2+r+1+\frac{13 r}{12} =0$
$\Rightarrow 12 r^2+25 r+12 =0$
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 $r=\frac{-4}{3}$.