The sum of the first six terms of an arithmetic progression is 54 and the ratio of its 10^th term to its 30^th term is 11 : 31. What is the sum of its first 15 terms?

Given:

The sum of the first six terms of an arithmetic progression is 54 and the ratio of its 10^th term to its 30^th term is 11:31.

Formula used:

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

S_n = n/2 × (2a + (n-1)d)

The n^th term of an AP is given by:

a_n = a + (n-1)d

Calculation:

Sum of the first six terms:

S₆ = 6/2 × (2a + 5d) = 54

⇒ 3 × (2a + 5d) = 54

⇒ 2a + 5d = 18 ... (i)

Ratio of 10^th term to 30^th term:

(a + 9d) / (a + 29d) = 11/31

⇒ 31(a + 9d) = 11(a + 29d)

⇒ 31a + 279d = 11a + 319d

⇒ 20a = 40d

⇒ a = 2d ... (ii)

Substitute a = 2d in equation (i):

2(2d) + 5d = 18

⇒ 4d + 5d = 18

⇒ 9d = 18

⇒ d = 2

Using a = 2d:

⇒ a = 2 × 2 = 4

Sum of the first 15 terms:

S₁₅ = 15/2 × (2a + 14d)

⇒ S₁₅ = 15/2 × (2 × 4 + 14 × 2)

⇒ S₁₅ = 15/2 × (8 + 28)

⇒ S₁₅ = 15/2 × 36

⇒ S₁₅ = 15 × 18

⇒ S₁₅ = 270

∴ The correct answer is option 4.