The first three terms of a geometric sequence are x,y,z and these have the sum equal to 42. If the middle term y is multiplied by 5/4 , the numbers now form an arithmetic sequences. The largest possible value of x is

The three terms of the geometric sequence with the common ratio r are x, xr, xr2
$\therefore \mathrm{x}+\mathrm{xr}+{\mathrm{xr}}^2= 42$
After multiplying the middle term by 5 4 . We will get an arithmetic sequence.
$\begin{array}{l}\therefore \frac{5}{4}\mathrm{xr}-\mathrm{x}={\mathrm{xr}}^2-\frac{5}{4}\mathrm{xr}\\ \therefore 2{\mathrm{r}}^2-5\mathrm{r}+2=0,\therefore \mathrm{r}=\frac{1}{2}\text{ and }\mathrm{r}=2\end{array}$
On substituting these in $\mathrm{x}+\mathrm{xr}+{\mathrm{xr}}^2=42$, we get $\mathrm{x}=6\text{ or }24$