Question

The sum of the 3rd term and 5th term of a g.p is -60 and the 5th term and 7th term is -240 find A the common ratio B the first term C sum of the 8th term

Answer 1

sum of the 3rd term and 5th term of a g.p is -60:
ar^2 + ar^4 = -60
ar^2(1 + r^2) = -60

the 5th term and 7th term is -240, (assuming you meant "sum")
ar^4 + ar^6 = -240
ar^4(1 + r^2) = -240

divide the 2nd by the first:
[ar^4(1 + r^2)] / [ar^2(1 + r^2) ] = -240/-60
r^2 = 4
r = ± 2

sub into the 1st:
ar^2(1+r^2) = -60
a(4)(5) = -60
a = -3

sum(8) = a(r^8 - 1)/(r-1)
= -3(255)/1 = -765 , if r = +2

sum(8) = -3(255)/-3 = 255 if r = -2

check:
if r = 2, we have
-3, -6, -12, -24, -48, -96, -192, -384, ... (all conditions are met)
if r = -2, we have
-3, 6, -12, 24, -48, 96, -192, 384, .... (all conditions are met)