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Maths Exercices

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) , , , , ... c) , , , , ...

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)

 

  , , , , ... is the same as:

, , , , ...

The general term is: bn=

c)  
  , , , , ...

The general term is: cn=