Question

Find three numbers $a,b,c$ which are $A.P$ between $2$ and $18$ such that their sum is $25$.

Answer 1

$2<a,b,c,<18$

$a+b+c=25$ ……. (1)

$2,a,b$ are AP

$\Rightarrow 2a=b+2$

$\Rightarrow 2a–b=2$ …….. (2)

$b,c,18$ are in GP

$\Rightarrow c^2=18b$ …… (3)

From (2) $\Rightarrow a=\frac{b+2}{2}$

$\Rightarrow b+\frac{2}{2}+b+c=25$

$\Rightarrow 3b=48–2c$

(3) $\Rightarrow c^2=6(48–2c)$

$\Rightarrow c^2+12c–288=0$

$\Rightarrow c=12,-24\left(rejected\right)$

$\Rightarrow a=5,b=8,c=12$