Question

Find $3$ terms in G.P. whose sum and product respectively are $7$ and $8$

Answer 1

Let the three terms be $\frac{a}{r},a,ar$.

It is given that the product of the terms is $8$, therefore,

$\frac{a}{r}\times a\times ar=8\Rightarrow a^3=8\Rightarrow a^3=2^3\Rightarrow a=2.....\left(1\right)$

It is also given that the sum of the terms is $7$, therefore using equation 1 we have,

$\frac{a}{r}+a+ar=7\Rightarrow \frac{2}{r}+2+2r=7\Rightarrow \frac{2}{r}+2r=7-2\Rightarrow 2(\frac{1}{r}+r)=5\Rightarrow \frac{1+r^2}{r}=\frac{5}{2}\Rightarrow 2r^2+2=5r\Rightarrow 2r^2-5r+2=0\Rightarrow 2r^2-4r-r+2=0\Rightarrow 2r(r-2)-1(r-2)=0\Rightarrow (2r-1)(r-2)=0\Rightarrow (2r-1)=0,(r-2)=0\Rightarrow r=\frac{1}{2},r=2$

With $a=2$ and $r=\frac{1}{2}$, the three terms of G.P are as follows:

$\frac{a}{r}=\frac{2}{\frac{1}{2}}=2\times 2=4$

$a=2$

$ar=2\times \frac{1}{2}=1$

Thus, the terms are $4,2$ and $1$.

With $a=2$ and $r=2$, the three terms are:

$\frac{a}{r}=\frac{2}{2}=1$

$a=2$

$ar=2\times 2=4$

Thus, the terms are $1,2$ and $4$.