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Arithmetic and geometric sequences

Introduction to sequences

A sequence is an ordered list of numbers:

3, 6, 9, 12, ....
1, 3, 5, 7, ...

A sequence can be thought of as a list of numbers written in a definite order:

a1, a2, a3, ..., an, ...

The number a1 is called the first term, a2 is the second term, and in general an is the nth term.

The terms of a sequence often follow a particular pattern. In those instances, we can determine the general term that expresses every term of the sequence. For example:

 Sequence General Term 3, 6, 9, 12,... 1,3,5,7,... 2,4,8,16,... 3n 2n-1 2n

Given an=3n+2, find a1, a2, a3 and a4.

By substituting n=1,2,3,4 in the general term of the sequence, we obtain:

a1=3(1)+2=3+2=5
a2=3(2)+2=6+2=8
a3=3(3)+2=9+2=11
a4=3(4)+2=12+2=14

Find the general term for the following sequences:

 a) 4, 8, 12, 16,... b) $\frac{4}{3}$, $\frac{4}{9}$, $\frac{4}{27}$, $\frac{4}{81}$, ... c) $\frac{1}{2}$, $\frac{2}{3}$, $\frac{3}{4}$, $\frac{4}{5}$, ...

Solution:

 a) 4, 8, 12, 16,... is the same as: 1·4, 2·4, 3·4, 4·4,... The general term is: an=4n b) $\frac{4}{3}$, $\frac{4}{9}$, $\frac{4}{27}$, $\frac{4}{81}$, ... is the same as: $\frac{4}{3}$, $\frac{4}{3^2}$, $\frac{4}{3^3}$, $\frac{4}{3^4}$, ... The general term is: bn= $\frac{4}{3^n}$ c) $\frac{1}{2}$, $\frac{2}{3}$, $\frac{3}{4}$, $\frac{4}{5}$, ... The general term is: cn=$\frac{n}{n+1}$