Sequences

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课程笔记

A sequence is a list of numbers that follows a rule. Each number is a term; u_n or T(n) is the nth term.
A term-to-term rule says how to get from one term to the next (e.g. add 4). A position-to-term rule (the nth term) gives any term directly.
Here positions are positive integers starting at n = 1.

An arithmetic (linear) sequence goes up or down by a constant common difference d.
The nth term is dn + (a - d), where a is the first term. For 5, 8, 11, 14: d = 3, so 3n + 2.
To check whether a number is in the sequence, solve 3n + 2 = \text{number} and see if n is a positive integer.

A geometric sequence multiplies by a constant common ratio r: 3, 6, 12, 24 has r = 2. The nth term is ar^n-1.
Fibonacci-type sequences add the previous two terms.

A constant nonzero second difference identifies a quadratic sequence an² + bn + c, with a equal to half that difference. Zero second difference gives a linear or constant sequence.
For 2, 7, 14, 23: first differences 5, 7, 9; second difference 2, so a = 1. Subtract n² (1, 4, 9, 16) to leave 1, 3, 5, 7 = 2n - 1. So n² + 2n - 1.

Square numbers 1, 4, 9, 16; cube numbers 1, 8, 27, 64; triangular numbers 1, 3, 6, 10; powers of 2: 2, 4, 8, 16.

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