Question

Find three numbers $a,b,c$ between $2$ and $18$ such that
(a) their sum is $25$
(b) the numbers $2,b,c$ are consecutive terms of an $A.P.$ and
(c) the number $b,c,18$ are consecutive terms of a $G.P$.

Answer 1

From the given relations, we have

$b+c+d=25,2b=2+c;d^2=18c.$

Eliminating b and c, we get

$2b+2c+2d=50$

or $(2+c)+2c+2d=50$

or $3c=48-2dor18c=6(48-2d)$

or $d^2=288-12d.$

$d^2+12d-288=0\therefore d=12$ (only+ive)

$\therefore c=8andb=5$