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: Understanding the Question:
We are given an Arithmetic Progression (A.P.): $-\frac{15}{4}, -\frac{10}{4}, -\frac{5}{4}, \dots$.
We need to find the value of the difference between the $16^{\text{th}}$ term and the $12^{\text{th}}$ term, represented as $a_{16} - a_{12}$.

Step 2: Key Formula or Approach:
The $n^{\text{th}}$ term of an Arithmetic Progression is given by:
\[ a_n = a + (n-1)d \]
where $a$ is the first term and $d$ is the common difference.
Using this formula, we can express the required difference as:
\[ a_{16} - a_{12} = [a + 15d] - [a + 11d] = 4d \]
This shows that the difference is independent of the first term $a$, and we only need to find the common difference $d$.

Step 3: Detailed Explanation:

• Identify the terms of the given A.P.:
First term, $a_1 = -\frac{15}{4}$
Second term, $a_2 = -\frac{10}{4}$

• Calculate the common difference $d$ by subtracting the first term from the second term:
\[ d = a_2 - a_1 \]
\[ d = -\frac{10}{4} - \left(-\frac{15}{4}\right) \]
\[ d = -\frac{10}{4} + \frac{15}{4} \]
\[ d = \frac{-10 + 15}{4} = \frac{5}{4} \]

• Use the simplified relationship for the difference of terms:
\[ a_{16} - a_{12} = 4d \]

• Substitute the value of $d = \frac{5}{4}$ into the equation:
\[ a_{16} - a_{12} = 4 \times \frac{5}{4} \]
\[ a_{16} - a_{12} = 5 \]


Step 4: Final Answer:
The value of $a_{16} - a_{12}$ is $5$.