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Write the explicit rule. Simplify your answer and do not include spaces.

Sequence: 32, 52, 72, 92, ...

[tex]\[ a_n = \][/tex]


Sagot :

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]
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