Question

Find three numbers in GP such that their sum is $14$ and the sum of their square is $84$-

• A. $3,6,12$
• B. $2,6,18$
• C. $1,3,9$
• D. $2,4,8$

Answer 1

The correct option is D $2,4,8$

Let three numbers in G.P$=a,ar,a{r}^2$

Sum of three number$=14$

$a+ar+a{r}^2=14$ ……..$\left(1\right)$

Sum of squares of number$=84$

$a^2+a^2{r}^2+a^2{r}^4=84$

Squaring equation $\left(1\right)$ both sides

$a^2+a^2{r}^2+a^2{r}^4+2a^{2}r+2a^2{r}^3+2a^2{r}^2=196$

$84+2a^{2}r(1+r^2+r)=196$

$2a^{2}r\frac{14}{a}=196-84$

$2ar14=112$

$a=\frac{112}{14\times 2r}=\frac{4}{r}$

$\frac{4}{r}+4+4r=14$

$4+4r+4r^2=14r$

$4r^2-10r+4=0$

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

$r=\frac{5\pm \sqrt{25-16}}{4}$

$r=\frac{5\pm 3}{4}$

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

$r=2,a=2$; $r=\frac{1}{2},a=8$

Three terms: $2,4,8$

Three terms: $8,4,2$

$\therefore$ Three terms are: $2,4,8$.