Not every sequence grows by a constant amount. Some sequences accelerate. Their first differences change, but the second differences stay constant. That pattern is a strong signal of a quadratic relationship.
This guide develops second differences as a diagnostic tool, connects them to quadratic nth terms, shows how to build a rule from data, and explains why a constant second difference is the sequence version of constant curvature in a quadratic graph.
Start with the differences
Consider:
2, 6, 12, 20, 30, ...
First differences: 4, 6, 8, 10
Second differences: 2, 2, 2The first differences are not constant, so the sequence is not arithmetic. The second differences are constant, suggesting a quadratic nth term.
Why quadratic sequences have constant second differences
Suppose Tn=an²+bn+c. The first difference is linear in n. Taking another difference removes that linear change and leaves a constant value of 2a.
Therefore:
constant second difference = 2aThis gives the coefficient of n² immediately.
Worked example 1: identify the quadratic coefficient
For the sequence 2, 6, 12, 20, 30, the constant second difference is 2.
2a=2
a=1So begin with n². The sequence n² gives 1, 4, 9, 16, 25. Compare with the target sequence: the differences are 1, 2, 3, 4, 5, which is n.
Tn=n²+nCheck n=5: 25+5=30.
Worked example 2: a more general quadratic
Find the nth term of 5, 12, 23, 38, 57, …
First differences: 7, 11, 15, 19
Second differences: 4, 4, 4Since 2a=4, a=2. Start with 2n²: 2, 8, 18, 32, 50. Subtract these from the sequence to get 3, 4, 5, 6, 7, which is n+2.
Tn=2n²+n+2A systematic coefficient method
For Tn=an²+bn+c, use the constant second difference to find a. Then subtract an² from the original sequence. The remainder should be linear, so find its nth term as bn+c.
This decomposition turns a difficult-looking rule into one quadratic layer plus one linear layer.
Worked example 3: reverse use
For Tn=n²+3n, find the 12th term.
T12=12²+3(12)=144+36=180Now test whether 70 belongs to the sequence:
n²+3n=70
n²+3n-70=0
(n+10)(n-7)=0The positive integer solution is n=7, so 70 is the 7th term.
Visual quadratic growth
Quadratic sequences often appear when a growing figure gains an increasing number of tiles each stage. For example, square numbers 1, 4, 9, 16, … have first differences 3, 5, 7, … and constant second difference 2.
The increasing first difference is visible in the border added to each larger square.
Connection to graphs
If term number n is plotted against Tn, a quadratic sequence lies on a parabola rather than a straight line. Constant first difference corresponds to constant gradient; constant second difference corresponds to a gradient that changes at a constant rate.
Common errors
Stopping after seeing changing first differences. Compute second differences before deciding the structure.
Using second difference as a directly. The second difference is 2a, so halve it.
Assuming every non-linear-looking sequence is quadratic. Constant second differences are the key clue.
Forgetting to test the final rule. Verify several terms, not only one.
Diagnostic table
| Observed mistake | Likely issue | Repair |
|---|---|---|
| Calls sequence arithmetic because differences “look patterned” | Difference order confusion | Check whether first differences are actually constant |
| Uses a=second difference | Quadratic coefficient rule | Use 2a=second difference |
| Finds n² part but cannot finish | Residual linear structure | Subtract an² and solve the remainder |
Practice
- Find the nth term of 3, 8, 15, 24, 35, …
- Find the nth term of 6, 15, 28, 45, 66, …
- For Tn=2n²-n, find T10.
- Determine whether 90 is in the sequence Tn=n²+2n.
Answers
1. n²+2n. 2. 2n²+3n+1. 3. 190. 4. n²+2n=90 gives n=9, so yes.
Connected routes
Use From Pattern Spotting to a General Term and Arithmetic Sequences as prerequisites, then continue to Recursive Rules and Explicit Rules. Return to the Mathematics Learning Hub.