Math Tools
Quadratic Sequence Calculator
Generate quadratic sequences, find nth terms, calculate second differences, and work out sums of the first n terms.
Quadratic Sequence Calculator
Generate quadratic sequence terms, find the nth term, second difference, and sum of the first n terms.
General rule
Tₙ = an² + bn + c
For a quadratic sequence, the second difference is constant.
Results
Nth term
36
Sum of first n terms
91
Second difference
2
Sequence
1, 4, 9, 16, 25, 36
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What this calculator does
A quadratic sequence has a constant second difference. This calculator uses the rule Tₙ = an² + bn + c to generate terms, find the nth term, and calculate the sum of the first n terms.
What is a quadratic sequence calculator?
A quadratic sequence calculator is a tool that helps you work with sequences whose nth term can be written in the form Tₙ = an² + bn + c. It can generate the first terms of the sequence, identify the constant second difference, calculate a chosen nth term, and find the sum of the first n terms.
This is useful for algebra students, teachers, exam practice, and anyone who wants to understand how quadratic number patterns grow. Instead of working everything out by hand, you can enter the coefficients and get the sequence and related values instantly.
Quadratic sequence formula and second difference
Nth term formula
Tₙ = an² + bn + c
Second difference rule
Second difference = 2a
In a quadratic sequence, the first differences are not constant, but the second differences are. That constant second difference is one of the quickest ways to recognize that a sequence is quadratic.
If the nth term is written as Tₙ = an² + bn + c, then the constant second difference is always 2a. This makes it easier to connect the sequence pattern to the coefficient of n².
Quadratic sequence examples table
These worked examples show the nth term rule, first terms, first differences, and constant second difference.
| Nth term rule | First terms | First differences | Second difference |
|---|---|---|---|
| Tₙ = n² | 1, 4, 9, 16, 25 | 3, 5, 7, 9 | 2 |
| Tₙ = n² + 2n | 3, 8, 15, 24, 35 | 5, 7, 9, 11 | 2 |
| Tₙ = 2n² + 3 | 5, 11, 21, 35, 53 | 6, 10, 14, 18 | 4 |
| Tₙ = 3n² - n + 2 | 4, 12, 26, 46, 72 | 8, 14, 20, 26 | 6 |
Generated terms and first differences
This table shows the generated quadratic sequence values for your inputs.
| n | Tₙ | First difference |
|---|---|---|
| 1 | 1 | — |
| 2 | 4 | 3 |
| 3 | 9 | 5 |
| 4 | 16 | 7 |
| 5 | 25 | 9 |
| 6 | 36 | 11 |
Constant second difference: 2
How to find the nth term of a quadratic sequence
To find the nth term of a quadratic sequence, start by checking that the second differences are constant. Once you know the sequence is quadratic, you can model it with the formula Tₙ = an² + bn + c.
After identifying the coefficients a, b, and c, substitute the term number n into the expression to find any required term. This calculator does that automatically and also shows the generated sequence and sum.
How to calculate the sum of the first n terms in a quadratic sequence
The sum of the first n terms tells you the total when the sequence values are added together from term 1 up to term n. This is useful in algebra problems, sequence analysis, and exam questions that ask for cumulative totals rather than just individual terms.
This quadratic sequence calculator works out that total directly from the coefficients a, b, and c and your chosen value of n, so you do not need to add each term manually.
Where quadratic sequences are useful
Quadratic sequences appear in algebra, graphing, pattern spotting, and mathematical modeling. They are common in school mathematics, especially when learning how number patterns connect to quadratic expressions and parabolas.
They are also useful for understanding growth patterns where changes increase in a structured way rather than at a constant rate. Seeing the nth term, first differences, and second differences together helps make those patterns easier to understand.
Quadratic sequence calculator FAQ and nth term questions
What is a quadratic sequence?
A quadratic sequence is a number pattern where the second difference is constant. Its nth term can usually be written as Tₙ = an² + bn + c.
How do you know if a sequence is quadratic?
Find the first differences between consecutive terms, then find the differences of those differences. If the second difference is constant, the sequence is quadratic.
What is the formula for the nth term of a quadratic sequence?
The general form is Tₙ = an² + bn + c, where a, b, and c are constants that define the sequence.
What is the second difference in a quadratic sequence?
The second difference is the difference between consecutive first differences. For Tₙ = an² + bn + c, the constant second difference is 2a.
Can this calculator generate terms and find the sum as well?
Yes. It can generate the quadratic sequence, find the nth term, calculate the constant second difference, and compute the sum of the first n terms.
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