Certainly! Let's find the [tex]\(19^{\text{th}}\)[/tex] term of the given arithmetic progression (A.P.): [tex]\(7, 13, 19, 25, \ldots\)[/tex].
In an arithmetic progression, each term after the first is obtained by adding a constant difference to the previous term. This constant is known as the common difference ([tex]\(d\)[/tex]).
First, identify the first term ([tex]\(a\)[/tex]) and the common difference ([tex]\(d\)[/tex]) of the A.P.:
- The first term, [tex]\(a\)[/tex], is [tex]\(7\)[/tex].
- The common difference, [tex]\(d\)[/tex], can be found by subtracting the first term from the second term:
[tex]\[
d = 13 - 7 = 6
\][/tex]
Now, we need to find the [tex]\(19^{\text{th}}\)[/tex] term of the A.P. The formula for the [tex]\(n^{\text{th}}\)[/tex] term ([tex]\(a_n\)[/tex]) of an A.P. is given by:
[tex]\[
a_n = a + (n - 1) \cdot d
\][/tex]
Substitute the values into the formula:
- [tex]\(a = 7\)[/tex]
- [tex]\(d = 6\)[/tex]
- [tex]\(n = 19\)[/tex]
Thus, we have:
[tex]\[
a_{19} = 7 + (19 - 1) \cdot 6
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
a_{19} = 7 + 18 \cdot 6
\][/tex]
Next, perform the multiplication:
[tex]\[
a_{19} = 7 + 108
\][/tex]
Finally, add the numbers together to find the [tex]\(19^{\text{th}}\)[/tex] term:
[tex]\[
a_{19} = 115
\][/tex]
Therefore, the [tex]\(19^{\text{th}}\)[/tex] term of the arithmetic progression is [tex]\(115\)[/tex].
