The sum of the first 6 terms of an arithmetic progression is 0 and its 4^th term is 2. If the sum of its first n terms is 1440, then the value of n is: 

Given:

The sum of the first 6 terms of an arithmetic progression is 0 and its 4^th term is 2. If the sum of its first n terms is 1440.

Formula used:

Sum of first n terms of an AP: \(S_n = \dfrac{n}{2} (2a + (n-1)d) \)

n^th term of an AP: \(a_n = a + (n-1)d \)

Calculation:

Given sum of first 6 terms:

\(\dfrac{6}{2} (2a + 5d) = 0 \)

⇒ \(3(2a + 5d) = 0\)

⇒ \(2a + 5d = 0\)

Given 4^th term:

\(a + 3d = 2 \)

Solving these two equations:

From \(2a + 5d = 0\)

⇒ \(a = -\dfrac{5d}{2}\)

Substitute in \(a + 3d = 2\)

⇒ \(-\dfrac{5d}{2} + 3d = 2\) ⇒ d = 4

Substitute d back in \(a = -\dfrac{5d}{2}\)

⇒ \(a = -\dfrac{5 \times 4}{2}\) ⇒ \(a = -10\)

Given sum of first n terms is 1440:

\(1440 = \dfrac{n}{2}(2a + (n-1)d)\)

⇒ \(2880 = n(-20 + 4n - 4)\)

⇒ \(4n^2 - 24n - 2880 = 0 \)

⇒ \(n^2 - 6n - 720 = 0\)

⇒ \(n^2 - 30n + 24n - 720 = 0\)

⇒ n(n - 30) + 24(n - 30) = 0

⇒ (n - 30) (n + 24) = 0

⇒ n = 30 or -24

Since n must be positive:

∴ The correct answer is option (2).