Question

Find three numbers a, b, c between 2 & 18 such that
(i) this sum is 25 are in GP
(ii) 2, a, b are in AP end
(iii) b, c, 18 are in GP

Answer 1

Given,

$a+b+c=25$..........(1)

$2,a,b$ are in A.P.

$\Rightarrow 2a=b+2$...........(2)

$b,c,18$ are in G.P.

$c^2=18b$...........(3)

From (2)

$a=\frac{b+2}{2}$

From (1)

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

$\Rightarrow b=\frac{48-2c}{3}$

$\therefore c^2=18\left(\frac{48-2c}{3}\right)$

$c^2+12c-288=0$

$(c+24)(c-12)=0$

$\therefore c=12,-24$

$b=\frac{48-2\left(12\right)}{3}=8$

$a=\frac{8+2}{2}=5$

Therefore the numbers are $5,8,12$.