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!