To find the term [tex]\( n \)[/tex] in the arithmetic sequence such that [tex]\( a_n = 492 \)[/tex], follow these steps:
1. Identify the first term and common difference:
- The first term [tex]\( a_1 \)[/tex] is given as 9.
- The second term [tex]\( a_2 \)[/tex] is given as 16.
- Calculate the common difference [tex]\( d \)[/tex]:
[tex]\[
d = a_2 - a_1 = 16 - 9 = 7
\][/tex]
2. Write the general formula for the arithmetic sequence:
- The [tex]\( n \)[/tex]-th term [tex]\( a_n \)[/tex] of an arithmetic sequence can be given by the formula:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
3. Substitute the given values into the formula to find [tex]\( n \)[/tex]:
- Given [tex]\( a_n = 492 \)[/tex], substitute [tex]\( a_1 \)[/tex] and [tex]\( d \)[/tex] into the formula:
[tex]\[
492 = 9 + (n - 1) \cdot 7
\][/tex]
4. Solve for [tex]\( n \)[/tex]:
- First, isolate the term containing [tex]\( n \)[/tex]:
[tex]\[
492 - 9 = (n - 1) \cdot 7
\][/tex]
- Simplify the left-hand side:
[tex]\[
483 = (n - 1) \cdot 7
\][/tex]
- Divide both sides by 7 to solve for [tex]\( n - 1 \)[/tex]:
[tex]\[
(n - 1) = \frac{483}{7} = 69
\][/tex]
- Finally, add 1 to both sides to solve for [tex]\( n \)[/tex]:
[tex]\[
n = 69 + 1 = 70
\][/tex]
Therefore, the term number [tex]\( n \)[/tex] in the given arithmetic sequence such that [tex]\( a_n = 492 \)[/tex] is [tex]\( \boxed{70} \)[/tex].
