Question

Three numbers are in $AP$ and their sum is $15$. If $1,4$ and $19$ are added to these numbers respectively the resulting numbers are in $G.P$. Find the numbers.

Answer 1

$a,b,c\to A.P.$

$2b=a+c,a+b+c=15$

$\Rightarrow b=5,c=10-a$

$a+1,b+4,c+19\to G.P\Rightarrow a+1,a,29-a\to G.P$

$9^2=81=(a+1)(29-a)$

Solving we get $a=2/a=26$

Numbers are $3,5,7$ (OR) $26,5,-16$