Find the sum of the G.P.:

7/6, 7/36, 7/216, 7/1296, ... ,n terms.

Given:

The Geometric Progression (G.P.) is: 7/6, 7/36, 7/216, 7/1296, ... up to n terms.

Formula Used:

The sum of the first 'n' terms of a G.P. is given by S_n = a(1 - rⁿ) / (1 - r)

Where,

a = first term of the G.P.

r = common ratio of the G.P.

n = number of terms

Calculation:

From the given G.P.:

First term (a) = 7/6

To find the common ratio (r), divide the second term by the first term:

⇒ r = (7/36) ÷ (7/6)

⇒ r = (7/36) × (6/7)

⇒ r = 1/6

Now, apply the formula for the sum of 'n' terms (S_n):

⇒ S_n = (7/6) × (1 - (1/6)ⁿ) / (1 - 1/6)

⇒ S_n = (7/6) × (1 - (1/6)ⁿ) / (5/6)

⇒ S_n = (7/6) / (5/6) × (1 - (1/6)ⁿ)

⇒ S_n = (7/5) × (1 - (1/6)ⁿ)

The sum of the given G.P. for n terms is (7/5) × (1 - (1/6)ⁿ).