Question

In an A.P., if $a_{14} - a_8 = 24$, then the common difference of the A.P. is

• A. $6$
• B. $4$
• C. $\pm 4$
• D. $3$

Hint:

For any A.P., the difference between any two terms $a_p$ and $a_q$ is always given directly by:
\[ a_p - a_q = (p - q)d \]
Using this, we get $(14 - 8)d = 24 \implies 6d = 24 \implies d = 4$ in a single line!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given the difference between the $14^{\text{th}}$ term and the $8^{\text{th}}$ term of an Arithmetic Progression (A.P.) as 24.
We need to determine the common difference $d$ of this A.P.

Step 2: Key Formula or Approach:
The $n^{\text{th}}$ term of an A.P. is given by the formula:
\[ a_n = a + (n-1)d \]
where $a$ is the first term and $d$ is the common difference.
By expressing both $a_{14}$ and $a_8$ in terms of $a$ and $d$, we can simplify their difference to find $d$ directly.

Step 3: Detailed Explanation:

• Write down the algebraic expressions for $a_{14}$ and $a_8$:
\[ a_{14} = a + (14 - 1)d = a + 13d \]
\[ a_8 = a + (8 - 1)d = a + 7d \]

• Substitute these expressions into the given difference equation:
\[ a_{14} - a_8 = 24 \]
\[ (a + 13d) - (a + 7d) = 24 \]

• Simplify the expression:
\[ a + 13d - a - 7d = 24 \]
\[ 6d = 24 \]

• Solve for the common difference $d$:
\[ d = \frac{24}{6} = 4 \]


Step 4: Final Answer:
The common difference of the A.P. is $4$.