Linear and quadratic sequences
- A linear sequence has a common difference d, and nth term dn + (first term − d). 5, 8, 11, 14 has nth term 3n + 2.
- A quadratic sequence has a constant second difference. For an² + bn + c, the second difference is 2a.
- Method: find a from the second difference, subtract an² from each term, then find the linear nth term of what is left.
- Example: 3, 9, 19, 33. The second difference is 4, so a = 2. Subtracting 2n² (2, 8, 18, 32) leaves 1, 1, 1, 1. The nth term is 2n² + 1.
Using the nth term
- Find a term by substituting: the 10th term of n² + 2n is 100 + 20 = 120.
- To check whether a number is in the sequence, set the nth term equal to it and see if n is a positive whole number. 3n + 2 = 100 gives n = 98/3, so 100 isn't a term.
- To find which term has a value, solve an equation: n² − 3n + 7 = 25 gives n² − 3n − 18 = 0, (n − 6)(n + 3) = 0, so n = 6.
Limiting values
- As n gets very large (n → ∞), some sequences get closer and closer to a limiting value.
- Divide the top and bottom by n: (2n + 1)/(n + 3) = (2 + 1/n)/(1 + 3/n). As n → ∞, 1/n → 0, so the limit is 2.
- The limit is the ratio of the n coefficients: (3n − 1)/(6n + 5) → 3/6 = 1/2.
Key terms
- nth term
- A formula giving any term of a sequence from its position n.
- Common difference
- The fixed amount added each time in a linear sequence.
- Second difference
- The difference between consecutive first differences.
- Quadratic sequence
- A sequence with a constant second difference.
- Limiting value
- The value a sequence approaches as n becomes very large.