Step 1. The sequence that we have is:
[tex]-31,-35,-39,-43[/tex]This is an arithmetic sequence and we need to find the 22nd term.
Step 2. The formula that defines an arithmetic sequence is:
[tex]a_n=a_1+d(n-1)[/tex]where an is the value of the sequence for the nth term, a1 is the first term, and d is the difference between the terms.
In this case:
[tex]a_1=-31[/tex]Step 3. Finding d, the difference between the terms:
Thus, the value of d is -4:
[tex]d=-4[/tex]Step 4. Using the formula from step 2, we define a general equation for the sequence:
[tex]\boxed{a_n=-31-4\left(n-1\right)}[/tex]Using this equation, we can find any value of the sequence.
Step 5. Since we need to find the 22nd term, we take this value as the n value:
[tex]n=22[/tex]And substitute it into our equation:
[tex]a_{22}=-31-4\left(22-1\right)[/tex]Solving the operations:
[tex]\begin{gathered} a_{22}=-31-4(21) \\ a_{22}=-31-84 \\ \boxed{a_{22}=-115} \end{gathered}[/tex]Answer:
[tex]\boxed{a_{22}=-115}[/tex]
