What is the sum of the first 14 terms of an A.P whose 11^th and 7^th terms are 18.25 and 14.25 respectively?

Given:

11th term (T₁₁) = 18.25

7th term (T₇) = 14.25

Number of terms (n) = 14

Formula used:

n^th term of an A.P: T_n = a + (n-1)d

Sum of n terms of an A.P: S_n = n/2 × [2a + (n-1)d]

Calculations:

For T₁₁: 18.25 = a + 10d

⇒ a + 10d = 18.25  ...(1)

For T₇: 14.25 = a + 6d

⇒ a + 6d = 14.25  ...(2)

Subtracting (2) from (1):

⇒ (a + 10d) - (a + 6d) = 18.25 - 14.25

⇒ 4d = 4

⇒ d = 1

Substitute d = 1 into equation (2):

⇒ a + 6(1) = 14.25

⇒ a + 6 = 14.25

⇒ a = 8.25

Sum of the first 14 terms:

S₁₄ = 14/2 × [2a + (14 - 1)d]

⇒ S₁₄ = 7 × [2(8.25) + 13(1)]

⇒ S₁₄ = 7 × [16.5 + 13]

⇒ S₁₄ = 7 × 29.5

⇒ S₁₄ = 206.5

∴ The correct answer is option (3).