Question

The common difference of the A.P.: \( a_1, a_2, \ldots, a_m \) is 13 more than the common difference of the A.P.: \( b_1, b_2, \ldots, b_n \). If \( b_{31} = -277 \), \( b_{43} = -385 \) and \( a_{78} = 327 \), then \( a_1 \) is equal to:

• A. \(16\)
• B. \(19\)
• C. \(24\)
• D. \(21\)

Hint:

When two A.P.s are related through their common differences, always find the difference first before solving for terms.

Answer 1

The Correct Option is B

Concept: For an arithmetic progression: \[ a_n = a_1 + (n-1)d \] Using given terms, we can form equations to find the common difference and first term.
Step 1: Find the common difference of A.P. \( b_n \) \[ b_{31} = b_1 + 30d_b = -277 \] \[ b_{43} = b_1 + 42d_b = -385 \] Subtracting: \[ 12d_b = -108 \Rightarrow d_b = -9 \]
Step 2: Find the common difference of A.P. \( a_n \) Given: \[ d_a = d_b + 13 = -9 + 13 = 4 \]
Step 3: Use the given term of A.P. \( a_n \) \[ a_{78} = a_1 + 77d_a \] \[ 327 = a_1 + 77(4) \] \[ 327 = a_1 + 308 \Rightarrow a_1 = 19 \]