nth term calculator: linear and quadratic sequences

Example: 5, 8, 11, 14, …

1. The difference is always 3, so the rule starts $3n$.
2. Compare $3n$ (3, 6, 9, 12) with the sequence: each term is 2 more.
3. nth term $= 3n + 2$.

A decreasing sequence like 20, 17, 14, 11 has a difference of $-3$: its rule is $-3n + 23$ (or $23 - 3n$).

Example: 2, 5, 10, 17, …

1. First differences: 3, 5, 7. Second differences: 2, 2. Constant, so it's quadratic.
2. Halve the second difference: $a = 1$, so the rule starts $n^2$.
3. Subtract $n^2$ (1, 4, 9, 16): left with 1, 1, 1, 1.
4. nth term $= n^2 + 1$.

Another: 3, 8, 15, 24 gives $n^2 + 2n$.

• Any term: substitute $n$. For $3n + 2$, the 100th term is $302$.
• Is a number in the sequence? Set the rule equal to it and solve: $3n + 2 = 50$ gives $n = 16$, a whole number, so 50 is the 16th term. $3n + 2 = 60$ gives $n = 19.33$, so 60 isn't in the sequence.

Geometric sequences multiply by the same number each time (2, 6, 18, 54: ×3), so their differences aren't constant. Fibonacci-type sequences add the previous two terms. The calculator tells you if a sequence isn't linear or quadratic so you can check for these patterns.

• Writing $n + 3$ instead of $3n$ for a difference of 3.
• Forgetting to halve the second difference in a quadratic.
• Getting the sign of the constant wrong after comparing.

Practise with our nth term questions and our guide to sequences.

• Check your rule with the first two terms before moving on.
• Show the difference rows: they earn method marks.
• Quadratic nth terms are Higher tier at GCSE.