The sequence you've provided is:

\[ 17, 40, 61, 80, 97 \]

Let's analyze the differences between consecutive terms to identify any pattern:

\[
\begin{align*}
40 - 17 & = 23 \\
61 - 40 & = 21 \\
80 - 61 & = 19 \\
97 - 80 & = 17 \\
\end{align*}
\]

The differences are decreasing by 2 each time: 23, 21, 19, 17.

This suggests that the sequence might be a quadratic sequence. To confirm, let's assume the general form of the sequence is:

\[ a_n = an^2 + bn + c \]

Where \( n \) is the position of the term in the sequence.

For the first term \( (n = 1) \):

\[ 17 = a(1)^2 + b(1) + c \Rightarrow a + b + c = 17 \]

For the second term \( (n = 2) \):

\[ 40 = a(2)^2 + b(2) + c \Rightarrow 4a + 2b + c = 40 \]

For the third term \( (n = 3) \):

\[ 61 = a(3)^2 + b(3) + c \Rightarrow 9a + 3b + c = 61 \]

Now we solve this system of equations to find \( a \), \( b \), and \( c \):

1. \( a + b + c = 17 \)
2. \( 4a + 2b + c = 40 \)
3. \( 9a + 3b + c = 61 \)

Let's solve these equations.

1. Subtract the first equation from the second:

\[
(4a + 2b + c) - (a + b + c) = 40 - 17
\]
\[
3a + b = 23
\]

2. Subtract the second equation from the third:

\[
(9a + 3b + c) - (4a + 2b + c) = 61 - 40
\]
\[
5a + b = 21
\]

Now, subtract these two new equations:

\[
(5a + b) - (3a + b) = 21 - 23
\]
\[
2a = -2 \Rightarrow a = -1
\]

Substitute \( a = -1 \) into \( 3a + b = 23 \):

\[
3(-1) + b = 23 \Rightarrow -3 + b = 23 \Rightarrow b = 26
\]

Finally, substitute \( a = -1 \) and \( b = 26 \) into \( a + b + c = 17 \):

\[
-1 + 26 + c = 17 \Rightarrow 25 + c = 17 \Rightarrow c = -8
\]

So the general term for the sequence is:

\[ a_n = -n^2 + 26n - 8 \]

To find the next term \( a_6 \):

\[
a_6 = -(6)^2 + 26(6) - 8 = -36 + 156 - 8 = 112
\]

The next term in the sequence should be **112**.

Do you want details or have any questions? 

Here are some related questions:

1. What is the difference between arithmetic and quadratic sequences?
2. How do you derive a quadratic equation from a sequence?
3. How can you use the finite difference method to identify sequence types?
4. What are some real-world applications of quadratic sequences?
5. How do different types of sequences relate to polynomial functions?

**Tip:** When dealing with sequences, always check the first and second differences to identify whether the sequence is arithmetic, quadratic, or something more complex.

Discover how to analyze the quadratic sequence 17, 40, 61, 80, 97 using the finite difference method and solve for the next term. Learn the step-by-step process of deriving the quadratic equation and understanding the pattern. Ideal for Grades 9-11 students.

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