sequence: 100, 95, 90, 85, in what position does the term 18 appear?

The given sequence is: \(100, 95, 90, 85, \dots\).

This sequence is an arithmetic sequence because each term decreases by a constant difference. Let's find the common difference (\(d\)):

\[
d = 95 - 100 = -5
\]

The first term (\(a_1\)) is 100.

The general formula for the \(n\)-th term (\(a_n\)) of an arithmetic sequence is:

\[
a_n = a_1 + (n-1) \times d
\]

We need to find the position \(n\) at which the term 18 appears. Set \(a_n = 18\):

\[
18 = 100 + (n-1) \times (-5)
\]

Solve for \(n\):

\[
18 = 100 - 5(n-1)
\]

\[
18 = 100 - 5n + 5
\]

\[
18 = 105 - 5n
\]

\[
5n = 105 - 18
\]

\[
5n = 87
\]

\[
n = \frac{87}{5} = 17.4
\]

Since \(n\) must be an integer, there is no position in the sequence where the term is exactly 18. This means the term 18 does not appear in this arithmetic sequence.

Would you like further details or have any questions?

Here are five related questions you might consider:

1. What happens if you find a non-integer value for \(n\) in an arithmetic sequence?
2. How can you verify if a certain term is part of an arithmetic sequence?
3. What is the significance of the common difference in determining the nature of a sequence?
4. How would you determine the closest term to a given value in an arithmetic sequence?
5. How do you find the total number of terms in a finite arithmetic sequence?

**Tip:** When solving for the position of a term in an arithmetic sequence, a non-integer result for \(n\) means the term doesn't exist in the sequence.

Learn how to determine the position of term 18 in an arithmetic sequence using the arithmetic sequence formula. This problem involves identifying the common difference and calculating the term's position in the sequence. Suitable for students in Grades 9-12.

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