To determine the algebraic expression that represents the $n$-th integer in the sequence, we need to find a pattern or rule for how the terms change. Let's break it down step by step:

Step 1: Identify the common difference

The given sequence is:

$26, 17, 8, -1, -10, -19$

Find the difference between consecutive terms:

$17 - 26 = -9$ $8 - 17 = -9$ $-1 - 8 = -9$ $-10 - (-1) = -9$

The difference is constant ($-9$), so this sequence is arithmetic with a common difference of $-9$.

Step 2: General formula for an arithmetic sequence

The general formula for an arithmetic sequence is:

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

Where:

• $a_1$ is the first term of the sequence,
• $d$ is the common difference,
• $n$ is the position of the term.

Here:

• $a_1 = 26$,
• $d = -9$.

Substitute these values into the formula:

$a_n = 26 + (n-1)(-9)$

Step 3: Simplify the expression

Expand $(n-1)(-9)$:

$a_n = 26 - 9n + 9$

Combine like terms:

$a_n = -9n + 35$

Step 4: Match with the given options

The expression $a_n = -9n + 35$ matches option C.

Final Answer:

C. $-9n + 35$

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Would you like more details or have any questions? Here are 5 related questions to expand your understanding:

1. How do we identify if a sequence is arithmetic?
2. Can this process be used for sequences with non-constant differences?
3. What is the significance of the first term ($a_1$) in the formula?
4. How would the formula change if the common difference were positive?
5. Can you derive the formula for a geometric sequence?

Tip: Always check the first few terms with your derived formula to ensure accuracy.