Sequences

A sequence is an ordered list of numbers or objects arranged according to a rule.

A sequence of numbers is a function s: N → R

Each number of the sequence is called a term, and we represent it as a_i. We call general term of a sequence, a_n, the expression which represents any term of the sequence.

For example:

· 1, 2, 3, 4, 5, … a_n = n, then a₁₀₀ = 100

· 2, 4, 6, 8, 10, … a_n = 2n, then a₂₅ = 50

· 1, 4, 9, 16, 25, … a_n = n², then a₁₂ = 144

· 32, 16, 8, 4, 2, … a_n = 2^6-n, then a₁₀ = 2^-4 = 1/16

Some sequences are recurring, because we obtain each term from the previous one. For example:
· 1, 1, 2, 3, 5, 8, 13, … a_n = a_n-1 + a_n-2     a₁ = a₂ = 1

This is the Fibonacci sequence.

An arithmetic progression is such a sequence of numbers that the difference of any two successive members of the sequence is a constant. The constant is called difference of the progression.
Example: 3, 5, 7, 9, …. d = 2
General term: a_n = a₁ + (n-1)· d
In the example: a_n = 3 + (n-1)· 2 = 2n + 1

A geometric progression is such a sequence of numbers that the quotient of any two successive members of the sequence is a constant, which is called ratio of the progression.
Example: 3, 6, 12, 24 …. r = 2
General term: a_n = a1· r^n-1
In the example: a_n = 3 · 2^n-1

 

Exercise: find the general term and a₁₀ of these sequences:

a) 1, 3, 9, 27, 81, ...

b) 15, 12, 9, 6, 3, 0, ...

c) 1, -2, 3, -4, 5, ...

 

 

 

 

Solutions: a) a_n=3^n-1, a₁₀= 3⁹= 19683; b) a_n= 18- 3n, a₁₀=-12; c) (-1)ⁿ·n; a₁₀= 10;