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:

Avoid calculating individual terms like \(a_{16}\) and \(a_{12}\) separately. Directly use the property \(a_n - a_m = (n-m)d\) to save time in MCQs.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
An Arithmetic Progression (A.P.) is a sequence where the difference between consecutive terms is constant. This difference is called the common difference (\(d\)). Any term \(a_n\) can be written as \(a + (n-1)d\).
Step 2: Key Formula or Approach:
The difference between the \(p^{\text{th}}\) and \(q^{\text{th}}\) terms of an A.P. is:
\[ a_p - a_q = (p - q)d \]
Step 3: Detailed Explanation:
The given A.P. is \(-\frac{15}{4}, -\frac{10}{4}, -\frac{5}{4}, \dots\)
First term (\(a\)) = \(-\frac{15}{4}\)
Common difference (\(d\)) = \((-\frac{10}{4}) - (-\frac{15}{4}) = \frac{5}{4}\)
We need to find \(a_{16} - a_{12}\).
Using the formula:
\[ a_{16} - a_{12} = (16 - 12)d \]
\[ a_{16} - a_{12} = 4d \]
Substitute the value of \(d\):
\[ a_{16} - a_{12} = 4 \times \left(\frac{5}{4}\right) = 5 \]
Step 4: Final Answer:
The value of \(a_{16} - a_{12}\) is \(5\).