To determine the common difference of an arithmetic progression (A.P.) where the [tex]\(n^{\text{th}}\)[/tex] term is given by the formula [tex]\(a_n = 2n + 3\)[/tex], follow these steps:
1. Understand the general form for the [tex]\(n^{\text{th}}\)[/tex] term of an A.P.:
- The general form for the [tex]\(n^{\text{th}}\)[/tex] term of an arithmetic progression is typically written as [tex]\(a_n = a + (n-1)d\)[/tex], where [tex]\(a\)[/tex] is the first term and [tex]\(d\)[/tex] is the common difference.
2. Identify the given formula:
- The formula provided for the [tex]\(n^{\text{th}}\)[/tex] term of this specific A.P is [tex]\(a_n = 2n + 3\)[/tex].
3. Calculate the first term ([tex]\(a_1\)[/tex]) and the second term ([tex]\(a_2\)[/tex]):
- The first term ([tex]\(a_1\)[/tex]) is when [tex]\(n = 1\)[/tex]:
[tex]\[
a_1 = 2(1) + 3 = 2 + 3 = 5
\][/tex]
- The second term ([tex]\(a_2\)[/tex]) is when [tex]\(n = 2\)[/tex]:
[tex]\[
a_2 = 2(2) + 3 = 4 + 3 = 7
\][/tex]
4. Determine the common difference ([tex]\(d\)[/tex]):
- The common difference [tex]\(d\)[/tex] in an arithmetic progression is the difference between the successive terms. Therefore:
[tex]\[
d = a_2 - a_1
\][/tex]
- Substitute the values of [tex]\(a_2\)[/tex] and [tex]\(a_1\)[/tex]:
[tex]\[
d = 7 - 5 = 2
\][/tex]
Therefore, the common difference of the A.P. is [tex]\(\boxed{2}\)[/tex].
