The common difference of an arithmetic progression ([tex]AP[/tex]) is -4, and its [tex]10^{\text{th}}[/tex] term is -8. Determine the first term of the [tex]AP[/tex].

To find the first term ([tex]$a_1$[/tex]) of an arithmetic progression (AP) where the common difference ([tex]$d$[/tex]) is -4 and the 10th term ([tex]$a_{10}$[/tex]) is -8, we will use the formula for the [tex]$n$[/tex]-th term of an AP.

The formula for the [tex]$n$[/tex]-th term of an AP is:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]

Given:
- Common difference ([tex]$d$[/tex]) = -4
- [tex]$10^{\text{th}}$[/tex] term ([tex]$a_{10}$[/tex]) = -8
- [tex]$n$[/tex] = 10

We need to find the first term ([tex]$a_1$[/tex]).

Step-by-step solution:

1. Write down the formula for the [tex]$n$[/tex]-th term:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]

2. Substitute the given values into the formula:
[tex]\[ -8 = a_1 + (10 - 1) \cdot (-4) \][/tex]

3. Simplify the expression inside the parentheses:
[tex]\[ -8 = a_1 + 9 \cdot (-4) \][/tex]

4. Multiply [tex]$9$[/tex] by [tex]$-4$[/tex]:
[tex]\[ -8 = a_1 - 36 \][/tex]

5. Rearrange the equation to solve for [tex]$a_1$[/tex] by adding 36 to both sides:
[tex]\[ a_1 = -8 + 36 \][/tex]

6. Simplify the right side of the equation:
[tex]\[ a_1 = 28 \][/tex]

Therefore, the first term of the arithmetic progression is [tex]$28$[/tex].