Question

In an A.P., the $6^{\text{th}}$ term is $52$ and the $11^{\text{th}}$ term is $112$. Then the common difference is equal to:

• A. \( 4 \)
• B. \( 20 \)
• C. \( 12 \)
• D. \( 8 \)
• E. \( 6 \)

Hint:

A faster way to find \( d \) is using the formula: \( d = \frac{a_p - a_q}{p - q} \). Substituting our values: \( d = \frac{112 - 52}{11 - 6} = \frac{60}{5} = 12 \).

Answer 1

The Correct Option is C

Concept: The \( n^{th} \) term of an A.P. is given by \( a_n = a + (n-1)d \), where \( a \) is the first term and \( d \) is the common difference.

Step 1: Set up the linear equations for the given terms.
From the \( 6^{th} \) term: \[ a + 5d = 52 \quad \cdots (1) \] From the \( 11^{th} \) term: \[ a + 10d = 112 \quad \cdots (2) \]

Step 2: Subtract equation (1) from equation (2) to eliminate \( a \).
\[ (a + 10d) - (a + 5d) = 112 - 52 \] \[ 5d = 60 \]

Step 3: Solve for \( d \).
\[ d = \frac{60}{5} \] \[ d = 12 \]