Question

A- Three numbers whose sum is 30, are in AP. If
1,2,4 are added to them respectively, they will be in GP. Find the numbers.

Answer 1

Let the terms of A.P. be:

$a-d,a,a+d$

Given,

$a-d+a+a+d=30$

$3a=30$

$\Rightarrow a=10$

$a-d+1,a+2,a+d+4$ are in G.P.

$\Rightarrow 11-d,12,14+d$ are in G.P.

$\frac{12}{11-d}=\frac{14+d}{12}$

$144=(14+d)(11-d)$

$d^2+3d-10=0$

$(d+5)(d-2)=0$

$d=-5,2$

when $d=2,a=10$ the series is, $8,10,12$

when $d=-5,a=10$ the series is, $15,10,5$