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.