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: Use the direct term-difference shortcut.
For any A.P., the difference between two terms $a_p$ and $a_q$ can be written straightaway as $a_p - a_q = (p-q)d$, without separately expanding both terms first.
Step 2: Plug in the given term numbers.
Here $p = 14$ and $q = 8$, and we are told $a_{14} - a_8 = 24$. So $(14 - 8)d = 24$, which gives $6d = 24$.
Step 3: Solve for the common difference.
Dividing both sides by 6, $d = \frac{24}{6} = 4$.
\[ \boxed{d = 4} \]