Question

The first three terms of a geometric progression are $x,y,z$ and $x+y+z=42.$ If the middle term, $y$ is multiplied by $\frac{5}{4}$, the numbers $x,\frac{5y}{4},z$ now form an arithmetic progression. The largest possible value of $x,$ is

• A. 6
• B. 12
• C. 24
• D. 20

Answer 1

The correct option is C 24

The three terms of the geometric progression with the common ratio $r,$ are $x,xr,x{r}^2$
$\therefore x+xr+x{r}^2=42$
Multiplying the middle term by $\frac{5}{4}$ results in an arithmetic progression.
So, $x,\frac{5}{4}xr,x{r}^2$ are in A.P.
This yields $\frac{5}{4}xr-x=x{r}^2-\frac{5}{4}xr$
$\Rightarrow 2r^5-5r+2=0$
$\Rightarrow r=\frac{1}{2}$ or $r=2.$
On substituting these in $x+xr+x{r}^2=42$
We get, $x=6$ or 24