1.5: Sequences and Difference Equations

Sequences

A sequence is a (possibly infinite) list of numbers written in a specific order: \[a_1, \ \ a_2, \ \ a_3, \ \ a_4, \ \ \dots, \ \ a_n, \ \ \dots\]

Usually, a formula for $a_n$ is given, allowing you to derive all the terms. $n$ usually starts at 1 unless otherwise stated.

Find the first five terms of the following sequences:
  1. $a_n = \dfrac{n}{n + 1}, \ \ n\geq 1$
  2. $a_n = (-1)^{n-1}, \ \ n\geq 1$

Sequences can be thought as functions.

The input is position on the list. Output is the actual term. Therefore:

The graph of a sequence is the set \[\{(n, a_n) : n \geq 1\}\]
Graph the sequence $a_n = (-1)^{n-1}, \ \ n\geq 1$.

Above we were given a formula for $a_n$. We can also be asked to find $a_n$ given a sequence.

Find a formula for $a_n$ of the sequence \[\dfrac{3}{5}, \ \ -\dfrac{4}{25}, \ \ \dfrac{5}{125}, \ \ -\dfrac{6}{625}, \ \ \dfrac{7}{3125}, \ \ \dots \]
Find a formula for $a_n$ of the sequence \[-1, \ \ \dfrac{1}{3}, \ \ -\dfrac{1}{5}, \ \ \dfrac{1}{7}, \ \ -\dfrac{1}{9}, \ \ \dots \]