Question

Find four positive consecutive terms in a G.P such that their product is $16$ and having sum of second and third terms equal to $5$ .

Answer 1

Let the numbers be,

$\frac{a}{r^2},\frac{a}{r},ar,a{r}^2$

Given,

$\frac{a}{r^2}\times \frac{a}{r}\times ar\times a{r}^2=16$

$\Rightarrow a^4=16$

$\therefore a=2$

$\frac{a}{r}+ar=5$

$a+a{r}^2=5r$

$2+2r^2=5r$

$2r^2-5r+2=0$

$(r-2)(r-\frac{1}{2})=0$

$\therefore r=2,\frac{1}{2}$

when, $a=2,r=2$ series is, $\frac{1}{2},1,4,8$

$\therefore r=2,\frac{1}{2}$

when, $a=2,r=\frac{1}{2}$ series is, $8,4,1,\frac{1}{2}$