Question

If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to:

• A. \( -1200 \)
• B. \( -1080 \)
• C. \( -1020 \)
• D. \( -120 \)

Hint:

For arithmetic progressions, use sum formulas effectively to simplify and solve equations.

Answer 1

The Correct Option is B

Step 1: Expressing given conditions. Given first term \( a = 3 \), we know: \[ S_4 = \frac{1}{5} (S_8 - S_4). \] \[ \Rightarrow 5S_4 = S_8 - S_4. \] \[ \Rightarrow 6S_4 = S_8. \] Step 2: Finding the common difference \( d \). Using sum formulas: \[ 6 \times \frac{4}{2} [2 \times 3 + (4-1) d] = \frac{8}{2} [2 \times 3 + (8-1) d]. \] \[ 12 (6 + 3d) = 4 (6 + 7d). \] \[ 18 + 9d = 6 + 7d. \] \[ \Rightarrow d = -6. \] Step 3: Finding \( S_{20} \). \[ S_{20} = \frac{20}{2} [2 \times 3 + (20-1)(-6)]. \] \[ = 10 [6 - 114] = -1080. \]