Question

The first three terms of an arithmetic-geometric progression are $3,-1$ and $-1.$ The next term of the progression is

• A. $2$
• B. $-2$
• C. $\frac{- 5}{27}$
• D. $\frac{- 5}{9}$

Answer 1

Answer: (D)

Let the first four terms of the $A.G.P.$ be
$3,\left(3 + d\right)r,\left(3 + 2 d\right)r^{2},\left(3 + 3 d\right)r^{3}$
Hence, $\left(3 + d\right)r=-1$ and $\left(3 + 2 d\right)r^{2}=-1$
$\frac{\left(3 + d\right)^{2} r^{2}}{\left(3 + 2 d\right) r^{2}}=\frac{1}{- 1}$
$\Rightarrow 9+d^{2}+6d=-3-2d$
$\Rightarrow d^{2}+8d+12=0\Rightarrow d=-2,-6$
$r=\frac{- 1}{3 + d}\Rightarrow r=-1$ or $\frac{1}{3}$
(i) for $d=-2,r=-1;$ next term is
$\left(3 + 3 d\right)r^{3}=\left(3 - 6\right)\left(- 1\right)^{3}=3$
(ii) for $d=-6,r=\frac{1}{3};$ next term is
$\left(3 + 3 d\right)r^{3}=\left(3 - 18\right)\left(\frac{1}{3}\right)^{3}=\frac{- 15}{27}=\frac{- 5}{9}$
So, the next term is $3$ or $\frac{- 5}{9}$