Sequences
Sequences
- A sequence follows a rule. The term-to-term rule gives the next term from the one before.
- Linear (constant difference): the difference is the multiple of $n$. e.g. $2, 5, 8, 11, \dots$ goes up by 3 → $n$th term $3n - 1$.
- Quadratic (the differences change steadily): e.g. $2, 5, 10, 17, \dots$ → $n$th term $n^2 + 1$.
Practice
The nth term of 2, 5, 8, 11, … is 3n − 1. What is the 10th term?
3 × 10 − 1 = 30 − 1 = 29.
Practice
The nth term of 2, 5, 10, 17, … is n² + 1. What is the 5th term?
5² + 1 = 25 + 1 = 26.
Practice
For a linear sequence going up by 3 each time, the nth term contains:
A constant difference of 3 means the nth term is 3n adjusted by a constant.
You've got it
Key idea
- linear sequence: $n$th term = (common difference)$\times n$ + adjust → $2,5,8,11 \to 3n-1$
- quadratic: second difference $= 2\times$ the $n^2$ coefficient → $2,5,10,17 \to n^2 + 1$
- exponential sequences multiply by a fixed number each step