Find the third term of a geometric progression if the sum of the first three terms is equal to 12, and the sum of the first six terms is equal to (−84).

Given:

The sum of the first three terms = 12

The sum of the first six terms = (−84).

To find:

The third term of a geometric progression.

Solution:

The sum of first n term of a geometric progression is:

Where, a is the first term and r is the common ratio.

The sum of the first three terms is equal to 12, and the sum of the first six terms is equal to (−84).

              ...(i)

              ...(ii)

Divide (ii) by (i), we get

Taking cube root on both sides, we get

Putting in (i), we get

Divide both sides by 3.

The nth term of a geometric progression is:

Where, a is the first term and r is the common ratio.

Putting in the above formula, we get

Therefore, the third term of the geometric progression is 16.