Question

Consider an arithmetic progression where the sum of the first 5 terms is 85, and the sum of the next 5 terms is 235. Find the common difference of the AP.

• A. 5
• B. 6
• C. 15
• D. 20

Hint:

When finding sum of terms other than first \( n \) terms in an AP, use the formula for sum of terms from \( (n+1) \) to \( (n+m) \):
\[ S = \frac{m}{2} \left[2a + (2n + m - 1)d\right] \] to avoid mistakes.

Answer 1

The Correct Option is B

Let the first term be \( a \) and common difference be \( d \). Sum of first 5 terms: \[ S_5 = \frac{5}{2} [2a + (5-1)d] = \frac{5}{2} (2a + 4d) = 85 \] Multiply both sides by 2: \[ 5 (2a + 4d) = 170 \implies 2a + 4d = 34 \quad \quad (1) \] Sum of next 5 terms (terms 6 to 10) is given as 235: \[ S_{6-10} = 235 \] Sum of first 10 terms: \[ S_{10} = \frac{10}{2} [2a + (10-1)d] = 5 (2a + 9d) \] Sum of next 5 terms is: \[ S_{6-10} = S_{10} - S_5 = 235 \] Substitute \( S_{10} \) and \( S_5 \): \[ 5 (2a + 9d) - 85 = 235 \] \[ 5 (2a + 9d) = 235 + 85 = 320 \] Divide both sides by 5: \[ 2a + 9d = 64 \quad \quad (2) \] Subtract equation (1) from equation (2): \[ (2a + 9d) - (2a + 4d) = 64 - 34 \] \[ 5d = 30 \implies d = 6 \]