To find the [tex]$25^{\text{th}}$[/tex] term of the arithmetic progression (AP) with the first term [tex]$a = 10$[/tex] and the common difference [tex]$d = -2$[/tex], we will use the formula for the [tex]$n^{\text{th}}$[/tex] term of an AP:
[tex]\[
a_n = a + (n-1) \cdot d
\][/tex]
Here, we need to find the [tex]$25^{\text{th}}$[/tex] term ([tex]$a_{25}$[/tex]).
Let's plug in the values:
- [tex]$a = 10$[/tex] (the first term)
- [tex]$d = -2$[/tex] (the common difference)
- [tex]$n = 25$[/tex] (we are looking for the 25th term)
Substitute these values into the formula:
[tex]\[
a_{25} = 10 + (25 - 1) \cdot (-2)
\][/tex]
First, simplify the expression inside the parentheses:
[tex]\[
25 - 1 = 24
\][/tex]
Then multiply this by the common difference [tex]$d$[/tex]:
[tex]\[
24 \cdot (-2) = -48
\][/tex]
Now, add this result to the first term [tex]$a$[/tex]:
[tex]\[
a_{25} = 10 + (-48)
\][/tex]
Combine the terms:
[tex]\[
a_{25} = 10 - 48 = -38
\][/tex]
Therefore, the [tex]$25^{\text{th}}$[/tex] term of the arithmetic progression is:
[tex]\[
\boxed{-38}
\][/tex]
