Question

If $x,2x+2,3x+3$ are in G.P., then the fourth term is:

• A. $27$
• B. $-27$
• C. $13.5$
• D. $-13.5$

Answer 1

The correct option is D

$-13.5$

Explanation for the correct option:

Geometric Progression:

It is given that $x,2x+2,3x+3$ are in G.P. So,

$$\begin{gathered} {\left(2x+2\right)}^2=x\left(3x+3\right) \\ \Rightarrow 4x^2+8x+4=3x^2+3x \\ \Rightarrow x^2+5x+4=0 \\ \Rightarrow \left(x+1\right)\left(x+4\right)=0 \\ \Rightarrow x=-1\mathrm{or}-4 \end{gathered}$$

When $x=-1$, then the series will be $-1,0,0$, this is not in G.P.

When $x=-4$, then the series will be $-4,-6,-9$, this is in G.P, where $a=-4,andr=\frac{3}{2}$.

So, the fourth term is $=a{r}^3=-4{\left(\frac{3}{2}\right)}^3=-\frac{27}{2}=-13.5\left[\mathrm{nth}\mathrm{term}\mathrm{of}G.P={\mathrm{ar}}^{n-1}\right]$

Hence, option D is correct.