A computer engineer wrote a software program that assigns a value each time the program is run. The first 5 assigned values are listed below.

–2, 5, 12, 19, 26

Which formula did the engineer use to determine the assigned values, where *n* is the number of times the program was run?

The sequence of values given is:

\[
-2, 5, 12, 19, 26
\]

This sequence is arithmetic because the difference between consecutive terms is constant. To determine the formula, let's first find the common difference \( d \).

### Step 1: Find the Common Difference
Subtract the first term from the second term:

\[
d = 5 - (-2) = 7
\]

Verify by checking the difference between other consecutive terms:

\[
12 - 5 = 7, \quad 19 - 12 = 7, \quad 26 - 19 = 7
\]

So, the common difference \( d \) is 7.

### Step 2: Find the Formula for the Sequence
The formula for the \(n\)th term of an arithmetic sequence is:

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

where:
- \( a_1 \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.

Here:
- \( a_1 = -2 \)
- \( d = 7 \)

Substitute these values into the formula:

\[
a_n = -2 + (n-1) \cdot 7
\]

### Step 3: Simplify the Formula
Distribute and simplify:

\[
a_n = -2 + 7n - 7 = 7n - 9
\]

So, the formula the engineer used is:

\[
\boxed{a_n = 7n - 9}
\]

Would you like more details, or do you have any questions?

Here are some related questions you might find useful:
1. What is an arithmetic sequence, and how do you identify it?
2. How do you find the common difference in a sequence?
3. What is the general formula for the \(n\)th term in an arithmetic sequence?
4. How do you determine the first term of a sequence if only the formula is given?
5. Can this method be applied to non-arithmetic sequences?
6. How does the formula change if the sequence has a different common difference?
7. How can you check if a given sequence is arithmetic?
8. How can this method be used to find missing terms in a sequence?

**Tip:** For arithmetic sequences, once you have the first term and the common difference, you can easily generate the entire sequence using the general formula.

Explore arithmetic sequences and learn how to determine the nth term formula with an example sequence -2, 5, 12, 19, 26. This problem involves finding the common difference and deriving the formula for the nth term. Suitable for students in Grades 7-9.

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