Question

The sum of the first and third terms of a G.P.is 7/3 and the sum of the second and fourth term is 7/4.Fin the first and the common difference

Answer 1

Use your definitions:

a + ar^2 = 7/3
a( 1 + r^2) = 7/3 **

ar + ar^3 = 7/4
a(r + r^3) = 7/4 ***

divide *** by **
(r + r^3)/(1 + r^2) = (7/4) / (7/3) = 3/4

4r + 4r^3 = 3 + 3r^2
4r^3 - 3r^2 + 4r - 3 = 0
r^2(4r - 3) + 1(4r-3) = 0
(4r-3)(r^2 + 1) = 0
4r = 3 or r^2 = -1 , the last is not possible, so

r = 3/4
sub back into ** to find a

btw, in a GP we have a common ratio , not a common difference.

Answer 2

Thanks very helpful