The sum of the first 2 terms of a G.P is 28,but the sum of the third and fourth is 252.Find the second term of the G.P if the seventh term is 5103

Answer 1

a + ar = 28 ----> a(1 + r) = 28
ar^2 + ar^3 = 252 ---> ar^2(1 + r) = 252
divide the 2nd by the first equation:
ar^2(1 + r) / a(1 + r) = 252/28
r^2 = 9
r = ± 3
if r = 3, a + 3a = 28 , a = 7
if r = -3, a -3a = 28, a = -14

we were also told that the seventh term is 5103
ar^6 = 5103
for r=3 and a =7
term7 = 7(3^6) = 5103, so that checks out
for r=-3, a=-14
term7 = -14(-3)^6 = -10206, so we have to reject those values of a and r

so we know a = 7, r = 3
2nd term = ar = 21