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The sequence provided is: 
\[ 1, 3, 7, 13, 21 \]

To find the general formula for the \(n\)-th term of a quadratic sequence, we need to determine the quadratic expression, typically written as:
\[
a_n = an^2 + bn + c
\]
where \(a\), \(b\), and \(c\) are constants to be determined.

### Step 1: Find the first and second differences

1. First differences (difference between consecutive terms):
\[
3 - 1 = 2, \quad 7 - 3 = 4, \quad 13 - 7 = 6, \quad 21 - 13 = 8
\]
So, the first differences are:
\[ 2, 4, 6, 8 \]

2. Second differences (difference between consecutive first differences):
\[
4 - 2 = 2, \quad 6 - 4 = 2, \quad 8 - 6 = 2
\]
The second differences are constant, which confirms that this is a quadratic sequence.

### Step 2: Use the second difference to find \(a\)
For a quadratic sequence, the second difference is always equal to \(2a\). Therefore:
\[
2a = 2 \quad \Rightarrow \quad a = 1
\]

### Step 3: Set up the general formula
Now we know that the general term has the form:
\[
a_n = n^2 + bn + c
\]

### Step 4: Use known terms to solve for \(b\) and \(c\)
We can substitute values from the given sequence to form a system of equations. Using the first three terms of the sequence:

1. For \(n = 1\): \(a_1 = 1^2 + b(1) + c = 1\), which simplifies to:
\[
1 + b + c = 1 \quad \Rightarrow \quad b + c = 0 \quad \text{(Equation 1)}
\]

2. For \(n = 2\): \(a_2 = 2^2 + b(2) + c = 3\), which simplifies to:
\[
4 + 2b + c = 3 \quad \Rightarrow \quad 2b + c = -1 \quad \text{(Equation 2)}
\]

### Step 5: Solve the system of equations
From Equation 1, \(b + c = 0\), so \(c = -b\). Substituting this into Equation 2:
\[
2b - b = -1 \quad \Rightarrow \quad b = -1
\]
Substitute \(b = -1\) back into Equation 1:
\[
-1 + c = 0 \quad \Rightarrow \quad c = 1
\]

### Step 6: Final expression
Substitute \(a = 1\), \(b = -1\), and \(c = 1\) into the general form:
\[
a_n = n^2 - n + 1
\]

Thus, the expression for the \(n\)-th term of the sequence is:
\[
a_n = n^2 - n + 1
\]

Would you like further details on the process, or do you have any other questions?

### Related Questions:
1. How can I verify that \(a_n = n^2 - n + 1\) works for other terms in the sequence?
2. What is the difference between quadratic sequences and arithmetic sequences?
3. Can quadratic sequences be used to model real-world phenomena?
4. How would the formula change if the second differences were not constant?
5. What happens if I add a constant to a quadratic sequence—how does that affect the formula?

### Tip:
Quadratic sequences always have a constant second difference, which is key to recognizing them and deriving their formulas!

Here are the first 5 terms of a quadratic sequence: 1, 3, 7, 13, 21. Find an expression, in terms of n, for the nth term of this quadratic sequence.

This tutorial explains how to find the nth term of the quadratic sequence 1, 3, 7, 13, 21. Discover how to calculate the first and second differences, use the formula n^2 + bn + c, and solve for the constants. Suitable for students in Grades 9-10.

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