Let's analyze the given arithmetic sequence:
Sequence: 32, 52, 72, 92, ...
First, identify the first term of the sequence (often denoted as [tex]\( a_1 \)[/tex]):
[tex]\[ a_1 = 32 \][/tex]
Next, find the common difference (denoted as [tex]\( d \)[/tex]). This is the difference between consecutive terms:
[tex]\[ d = 52 - 32 = 20 \][/tex]
The general formula for the [tex]\( n \)[/tex]-th term of an arithmetic sequence is:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
Substitute the values of [tex]\( a_1 \)[/tex] and [tex]\( d \)[/tex] into the formula:
[tex]\[ a_n = 32 + (n - 1) \cdot 20 \][/tex]
Now, simplify the expression:
[tex]\[ a_n = 32 + 20n - 20 \][/tex]
[tex]\[ a_n = 20n + 12 \][/tex]
Thus, the explicit rule for the [tex]\( n \)[/tex]-th term of the sequence is:
[tex]\[ a_n = 20n + 12 \][/tex]
