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------------------------------------------------ The first 5 numbers of a sequence are shown below. Which of the following functions produces the sequence for [tex]n: \{1, 2, \ldots, n\}[/tex]?

[tex]26, 35, 44, 53, 62, \ldots[/tex]

A. [tex]f(n) = 17 - 9n[/tex]

B. [tex]f(n) = 9n + 26[/tex]

C. [tex]f(n) = 9n + 17[/tex]

D. [tex]f(n) = 26 - 9n[/tex]

To determine which function produces the given sequence [tex]$26, 35, 44, 53, 62, \ldots$[/tex], let's go through each function option provided:

The sequence provided is:

1. [tex]$ 26 $[/tex]
2. [tex]$ 35 $[/tex]
3. [tex]$ 44 $[/tex]
4. [tex]$ 53 $[/tex]
5. [tex]$ 62 $[/tex]

Let's evaluate each function option:

- Option A: [tex]$ f(n) = 17 - 9n $[/tex]
Substituting the values of [tex]$ n = 1, 2, 3, 4, 5 $[/tex]:
- For [tex]$ n = 1 $[/tex]: [tex]$ f(1) = 17 - 9 \times 1 = 8 $[/tex]
- For [tex]$ n = 2 $[/tex]: [tex]$ f(2) = 17 - 9 \times 2 = -1 $[/tex]
- For [tex]$ n = 3 $[/tex]: [tex]$ f(3) = 17 - 9 \times 3 = -10 $[/tex]
- For [tex]$ n = 4 $[/tex]: [tex]$ f(4) = 17 - 9 \times 4 = -19 $[/tex]
- For [tex]$ n = 5 $[/tex]: [tex]$ f(5) = 17 - 9 \times 5 = -28 $[/tex]

The results [tex]$8, -1, -10, -19, -28$[/tex] do not match the given sequence.

- Option B: [tex]$ f(n) = 9n + 26 $[/tex]
Substituting the values of [tex]$ n = 1, 2, 3, 4, 5 $[/tex]:
- For [tex]$ n = 1 $[/tex]: [tex]$ f(1) = 9 \times 1 + 26 = 35 $[/tex]
- For [tex]$ n = 2 $[/tex]: [tex]$ f(2) = 9 \times 2 + 26 = 44 $[/tex]
- For [tex]$ n = 3 $[/tex]: [tex]$ f(3) = 9 \times 3 + 26 = 53 $[/tex]
- For [tex]$ n = 4 $[/tex]: [tex]$ f(4) = 9 \times 4 + 26 = 62 $[/tex]
- For [tex]$ n = 5 $[/tex]: [tex]$ f(5) = 9 \times 5 + 26 = 71 $[/tex]

The sequence [tex]$35, 44, 53, 62, 71$[/tex] does not match the given sequence starting from the first term.

- Option C: [tex]$ f(n) = 9n + 17 $[/tex]
Substituting the values of [tex]$ n = 1, 2, 3, 4, 5 $[/tex]:
- For [tex]$ n = 1 $[/tex]: [tex]$ f(1) = 9 \times 1 + 17 = 26 $[/tex]
- For [tex]$ n = 2 $[/tex]: [tex]$ f(2) = 9 \times 2 + 17 = 35 $[/tex]
- For [tex]$ n = 3 $[/tex]: [tex]$ f(3) = 9 \times 3 + 17 = 44 $[/tex]
- For [tex]$ n = 4 $[/tex]: [tex]$ f(4) = 9 \times 4 + 17 = 53 $[/tex]
- For [tex]$ n = 5 $[/tex]: [tex]$ f(5) = 9 \times 5 + 17 = 62 $[/tex]

The results [tex]$26, 35, 44, 53, 62$[/tex] match the given sequence perfectly.

- Option D: [tex]$ f(n) = 26 - 9n $[/tex]
Substituting the values of [tex]$ n = 1, 2, 3, 4, 5 $[/tex]:
- For [tex]$ n = 1 $[/tex]: [tex]$ f(1) = 26 - 9 \times 1 = 17 $[/tex]
- For [tex]$ n = 2 $[/tex]: [tex]$ f(2) = 26 - 9 \times 2 = 8 $[/tex]
- For [tex]$ n = 3 $[/tex]: [tex]$ f(3) = 26 - 9 \times 3 = -1 $[/tex]
- For [tex]$ n = 4 $[/tex]: [tex]$ f(4) = 26 - 9 \times 4 = -10 $[/tex]
- For [tex]$ n = 5 $[/tex]: [tex]$ f(5) = 26 - 9 \times 5 = -19 $[/tex]

The results [tex]$17, 8, -1, -10, -19$[/tex] do not match the given sequence.

Therefore, the function that produces the sequence [tex]$26, 35, 44, 53, 62, \ldots$[/tex] is Option C: [tex]$ f(n) = 9n + 17 $[/tex].