Correct Answer - Option 2 :
\(-\frac{64}{3}\)
Concept:
Geometric Progression (GP): The series of numbers where the ratio of any two consecutive terms is the same, is called a Geometric Progression.
A Geometric Progression of n terms with first term a and common ratio r is represented as:
a, ar, ar2, ar3, ..., arn-2, arn-1.
Calculation:
Given: Sum of the first two terms of GP is -2 and the fifth term of GP is 4 times the third term of GP
According to the question:
a + ar = -2 ...(1)
ar4 = 4 × ar2
r = ±2 ...(2)
Using equation (1), we get:
a = \(-\frac{2}{3}\)
Now, the sixth term of the GP = ar5 = \(-\frac{64}{3}\).