In an A.P., S_n = 5, a = 5/7 and d = -1/21. Find the value of n?

Given:

Sum of the first n terms, S_n = 5

First term, a = 5/7

Common difference, d = -1/21

Concept Used:

The sum of the first n terms of an arithmetic progression (A.P.) is given by:

Formula: S_n = n/2 × [2a + (n-1)d]

Where:

n = number of terms

a = first term

d = common difference

Calculation:

Using the formula S_n = n/2 × [2a + (n-1)d]:

Substituting the given values:

5 = n/2 × [2 × (5/7) + (n-1) × (-1/21)]

⇒ 5 = n/2 × [(10/7) + (-n/21 + 1/21)]

⇒ 5 = n/2 × [(10/7) + (1/21 - n/21)]

⇒ 5 = n/2 × [(30/21) + (1/21 - n/21)]

⇒ 5 = n/2 × [(31/21) - (n/21)]

⇒ 5 = n/2 × [(31 - n)/21]

⇒ 5 = n × (31 - n) / 42

⇒ 210 = n × (31 - n)

⇒ 210 = 31n - n²

Rearranging:

⇒ n² - 31n + 210 = 0

This is a quadratic equation.

Solve using the quadratic formula:

n = [-b ± √(b² - 4ac)] / 2a

Here: a = 1 , b = -31 , c = 210

Substituting:

⇒ n = [-(-31) ± √((-31)² - 4 × 1 × 210)] / 2 × 1

⇒ n = [31 ± √(961 - 840)] / 2

⇒ n = [31 ± √121] / 2

⇒ n = [31 ± 11] / 2

Two possible values for n:

1. n = (31 + 11) / 2 = 42 / 2 = 21

2. n = (31 - 11) / 2 = 20 / 2 = 10

Conclusion:

Both n = 21 and n = 10 satisfy the equation.

Therefore, the correct answer is  option 1.
∴ n = 21, 10