To find the 500th term of the arithmetic sequence given by [tex]\( 24, 30, 36, 42, 48, \ldots \)[/tex], we will use the explicit formula for the [tex]\( n \)[/tex]-th term of an arithmetic sequence:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
where:
- [tex]\( a_n \)[/tex] is the [tex]\( n \)[/tex]-th term,
- [tex]\( a_1 \)[/tex] is the first term,
- [tex]\( d \)[/tex] is the common difference,
- [tex]\( n \)[/tex] is the number of the term.
Given:
- First term ([tex]\( a_1 \)[/tex]) is 24,
- Common difference ([tex]\( d \)[/tex]) is the difference between any two consecutive terms. Here, [tex]\( d = 30 - 24 = 6 \)[/tex],
- [tex]\( n = 500 \)[/tex] (we need to find the 500th term).
Substituting these values into the formula, we have:
[tex]\[
a_{500} = 24 + (500 - 1) \cdot 6
\][/tex]
Now, calculate step-by-step:
[tex]\[
a_{500} = 24 + (499) \cdot 6
\][/tex]
[tex]\[
a_{500} = 24 + 2994
\][/tex]
[tex]\[
a_{500} = 3018
\][/tex]
Therefore, the 500th term of the sequence is [tex]\( 3018 \)[/tex].
Comparing this result with the given answer choices:
A. 2994
B. 3018
C. 3024
D. 3042
The correct answer is B. [tex]\( 3018 \)[/tex].
