Question

Three numbers whose sum is 15 are in A.P. If 1,4,19 be added to them respectively, then they are in G.P. How many triplet sets of such numbers are possible ?

Answer 1

Let three numbers in A.P. are $a-d,a,a+d.$

$\therefore$ Sum $=3a=15\therefore a=5$

$\therefore$The numbers are $5-d,5,5+d.$

Adding $1,4,19$ respectively to these, they become $6-d,9,24+d,$

and these are given to be in G.P.$\therefore 9^2=(6-d)(24+d)$ or $d^2+18d-63=0(d+21)(d-3)=0$

$\therefore$ $d=-21$ or $3.$

Hence the numbers are $26,5,-16$ or $2,5,8$.