Question

For some positive real number '$b$'. the first $3$ terms in a geometric sequence (progression) are $b-1,b+4,3b+2$. What is the numerical value of the fourth term?

• A. $16$
• B. $20$
• C. $24$
• D. $28$
• E. $40$

Answer 1

The correct option is E $40$

let the first term be $a$ and geometric ratio be $r$
we have $a=b-1$ , $ar=b+4$ which gives $r=\frac{(b+4)}{(b-1)}$ and $a{r}^2=3b+2$ which implies $\frac{(b-1){(b+4)}^2}{{(b-1)}^2}=\frac{{(b+4)}^2}{(b-1)}=3b+2$
we get $b^2+8b+16=3b^2-b-2$ , which gives $2b^2-9b-18=0$
$b=6,-\frac{3}{2}$ , given $b$ is positive . So $b=6$
We get $a=6-1=5$ and $r=\frac{10}{5}=2$
Therefore fourth term is $a{r}^3=5\times 8=40$