Question

If $T_1,T_2,T_7$ of an A.P. consitute a G.P. whose sum is 93, then find the smallest of the numbers ?

Answer 1

$a,a+d,a+6d$ are in G.P

$(a+d{)}^2=a(a+6d)$

$d^2-4ad=0$

$d(d-4a)=0$

$d=0$ or $d=4a$

Since d cannot be zero , therefore we take $d=4a$

Numbers are $a,5a,25a$ where $a+5a+25a=31a=93$ $a=3,d=12$

Hence the numbers are $3,15,75$ which are clearly in G.P.