Let's analyze the problem step by step.

The sequence given is: $2, 5, 10, 17, \ldots$.

We need to find the explicit expression for the $n$-th term. First, let's observe the pattern in the sequence:

• 2nd term: $5 - 2 = 3$
• 3rd term: $10 - 5 = 5$
• 4th term: $17 - 10 = 7$

The differences between consecutive terms are increasing by 2 each time, which suggests that the sequence follows a quadratic pattern.

Now let's compare the given functions to see which one correctly describes this sequence:

1. Austin's function: $(n+1)^2 + 1$

   • For $n=1$: $(1+1)^2 + 1 = 2^2 + 1 = 4 + 1 = 5$ (but the first term is 2)
   • This function does not match the first term of the sequence.

2. Kaylee's function: $3n - 1$

   • For $n=1$: $3(1) - 1 = 3 - 1 = 2$
   • For $n=2$: $3(2) - 1 = 6 - 1 = 5$
   • For $n=3$: $3(3) - 1 = 9 - 1 = 8$ (but the third term is 10)
   • This function does not match.

3. Noah's function: $n^2 + 1$

   • For $n=1$: $1^2 + 1 = 1 + 1 = 2$
   • For $n=2$: $2^2 + 1 = 4 + 1 = 5$
   • For $n=3$: $3^2 + 1 = 9 + 1 = 10$
   • For $n=4$: $4^2 + 1 = 16 + 1 = 17$
   • This matches the sequence exactly!

4. Zoey's function: $7n - 11$

   • For $n=1$: $7(1) - 11 = 7 - 11 = -4$
   • This function does not match the sequence.

Thus, Noah wrote the correct expression, which is $n^2 + 1$.

Would you like a detailed explanation of the quadratic sequence patterns or have further questions? Here are some related questions for further exploration:

1. How can you find the pattern in a quadratic sequence?
2. What is the general form of a quadratic sequence?
3. How do you derive an explicit formula for a sequence from its terms?
4. How do linear and quadratic sequences differ?
5. Can a cubic function ever describe a quadratic sequence?

Tip: Always check the differences between terms to identify the degree of the sequence!