Sequences, Series and Summations

-- Section 8.2 --

 
Sequence: a function whose domain is the set of natural numbers.
natural numbers (counting numbers) {1, 2, 3, 4...}
Series: is a sum of a sequence.

  Write the first five terms of the sequence.   an = 2n2 - n

Solve

            Step

   

a1 = 2(1)2 - 1 = 2(1) - 1 = 2 - 1 = 1

Start with n = 1
a2 = 2(2)2 - 2 = 2(4) - 2 = 8 - 2 = 6  n = 2
a3 = 2(3)2 - 3 = 2(9) - 3 = 18 - 3 = 15 n = 3
a4 = 2(4)2 - 4 = 2(16) - 4 = 32 - 4 = 28 n = 4
a5 = 2(5)2 - 5 = 2(25) - 5 = 50 - 5 = 45  n = 5

The first five terms are 1, 6, 15, 28, 45

an = 2n2 - can also be written as  f(n) = 2n2 - n .  The term an indicates it is a special function called a sequence.

  
  Write the first five terms of the sequence.   an = n2 + 3n

[Solution]

  Evaluate the sum. 

We add the first five terms of the sequence an = 2n2 - n   from example 1.  

Solve

            Step

The first five terms are 1, 6, 15, 28, 45

=  1 + 6 + 15 + 28 + 45 = 95 n = 1 means start at 1
5 means
end at 5.

 

 Evaluate the sum. 

[Solution]

Evaluate the sum.

Solve

            Step

  
K = 3 means start at 3
5 means
end at 5.

k = 3, 4, 5

Add the fractions
The sum of a sequence is called a series.

Evaluate the sum.  

[Solution]

Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada