To find the common difference of an arithmetic progression (A.P.) given that [tex]\(a_{15} - a_{11} = 48\)[/tex], follow these steps:
1. Recall the formula for the [tex]\(n\)[/tex]-th term of an arithmetic progression:
[tex]\[
a_n = a + (n-1)d
\][/tex]
where [tex]\(a\)[/tex] is the first term, [tex]\(d\)[/tex] is the common difference, and [tex]\(n\)[/tex] is the term number.
2. For the 15th term ([tex]\(a_{15}\)[/tex]):
[tex]\[
a_{15} = a + 14d
\][/tex]
3. For the 11th term ([tex]\(a_{11}\)[/tex]):
[tex]\[
a_{11} = a + 10d
\][/tex]
4. According to the given information, the difference between the 15th term and the 11th term is 48:
[tex]\[
a_{15} - a_{11} = 48
\][/tex]
5. Substitute the expressions for [tex]\(a_{15}\)[/tex] and [tex]\(a_{11}\)[/tex] into the equation:
[tex]\[
(a + 14d) - (a + 10d) = 48
\][/tex]
6. Simplify the equation:
[tex]\[
a + 14d - a - 10d = 48
\][/tex]
7. Combine like terms:
[tex]\[
14d - 10d = 48
\][/tex]
[tex]\[
4d = 48
\][/tex]
8. Solve for [tex]\(d\)[/tex]:
[tex]\[
d = \frac{48}{4}
\][/tex]
[tex]\[
d = 12
\][/tex]
Thus, the common difference [tex]\(d\)[/tex] of the arithmetic progression is 12.
So the correct answer is:
(A) 12
