Question

If $a_n$ represents $n^{\text{th}}$ term of the A.P. $-\frac{15}{4}, -\frac{10}{4}, -\frac{5}{4}, \dots$ then value of $a_{16} - a_{12}$ is

• A. $4$
• B. $\frac{5}{4}$
• C. $5$
• D. $\frac{25}{4}$

Hint:

For any A.P., the difference between any two terms $a_p$ and $a_q$ is always given directly by $(p - q)d$.
Here, $a_{16} - a_{12} = (16 - 12)d = 4d$. You do not need to calculate the actual values of $a_{16}$ and $a_{12}$!

Answer 1

The Correct Option is C

Step 1: Use the shortcut for the difference of two terms.
For any A.P., $a_p - a_q = (p-q)d$, so we don't need to find $a$ at all. Here $a_{16}-a_{12} = 4d$.
Step 2: Find the common difference from the given terms.
The A.P. is $-\frac{15}{4}, -\frac{10}{4}, -\frac{5}{4}, \dots$, so
\[ d = -\frac{10}{4} - \left(-\frac{15}{4}\right) = \frac{5}{4} \]
Step 3: Substitute into the shortcut.
\[ a_{16}-a_{12} = 4 \times \frac{5}{4} = 5 \]
Step 4: Confirm the answer.
This matches option (3).
\[ \boxed{5} \]