Question

The value of \( a = 3 \), the sum of first 4 terms is equal to the one fifth of the sum of next 4 terms in A.P., then find the sum of the first 20 terms.

• A. 340
• B. 360
• C. 380
• D. 400

Hint:

In A.P. problems, always use the formula for the sum of terms and apply the given conditions to solve for the unknown common difference or terms.

Answer 1

The Correct Option is B

Let the first term of the arithmetic progression (A.P.) be \( a = 3 \), and the common difference be \( d \). The sum of the first 4 terms is: \[ S_4 = 4a + 6d = 4(3) + 6d = 12 + 6d. \] The sum of the next 4 terms (terms 5 to 8) is: \[ S_8 - S_4 = (8a + 28d) - (4a + 6d) = 4a + 22d = 12 + 22d. \] It is given that: \[ S_4 = \frac{1}{5}(S_8 - S_4). \] Substitute the expressions for \( S_4 \) and \( S_8 - S_4 \): \[ 12 + 6d = \frac{1}{5}(12 + 22d). \] Solve for \( d \): \[ 5(12 + 6d) = 12 + 22d \quad \Rightarrow \quad 60 + 30d = 12 + 22d \quad \Rightarrow \quad 8d = -48 \quad \Rightarrow \quad d = -6. \] Now, find the sum of the first 20 terms: \[ S_{20} = \frac{20}{2} \left[ 2a + (n-1)d \right] = 10 \left[ 2(3) + 19(-6) \right] = 10 \left[ 6 - 114 \right] = 10 \times (-108) = 360. \] Thus, the correct answer is option (2) 360.