In a GP, the sum of the 3rd and 4th terms is -4 /3, and the sum of the 4th and 5th terms is -4 /9. Find the 6th term

Answer 1

ar^2 + ar^3 = -4/3
ar^2(1 + r) = -4/3 **

ar^3 + ar^4 = -4/9
ar^3(1 + r) = -4/9 ***

divide *** by **
ar^3(1+r)/(ar^2(1+r)) = (-4/9) / (-4/3)

r = 1/3

plug back into **
a(1/3)^2 (1 + 1/3) = -4/3
a(1/9)(4/3) = -4/3
a = -9

term(6) = ar^5 = -9(1/3)^5 = -1/27

check: sequence is:
-9, -3, -1, -1/3, -1/9, -1/27, ...

is -1 - 1/3 = -4/3 ? , YES
is -1/3 - 1/9 = -4/9 ? YES