Determine the Term-to-Term Rule for the Sequence: 33, 25, 17, 9

To solve this problem, we'll analyze the sequence to find the common difference, determine the general form of the sequence formula, and then verify which of the given multiple-choice options corresponds to our result.

• Step 1: Calculate the common difference $d$.
  The sequence is: 33, 25, 17, 9,...
  Calculate differences between terms:
  $25 - 33 = -8$
  $17 - 25 = -8$
  $9 - 17 = -8$
  The common difference $d$ is $-8$.
• Step 2: Identify the first term $a_1$.
  The first term of the sequence is 33.
• Step 3: Write the formula for the $n$-th term of the sequence.
  Using the formula $a_n = a_1 + (n-1)d$, substitute $a_1 = 33$ and $d = -8$:
  $a_n = 33 + (n-1)(-8)$
  Simplifying, we have:
  $a_n = 33 - 8n + 8$
  $a_n = 41 - 8n$
• Step 4: Compare the formula $a_n = 41 - 8n$ with the answer choices to find the matching option.

The term-to-term rule for this sequence is $-8n + 41$, which corresponds to choice 2.