Tutorial on sequences and summations.
A  Introduction to Sequences
What is a sequence? A sequence may be thought of a list of ordered numbers. The sequence (or ordered list) may be finite or infinite.
Example 1:
2, 4, 6, 8, 10, ...
Example 2:
1/3, 1/5, 1/7, 1/9, 1/11, ...
We use subscripts to write the terms of a sequence as follows
a_{ 1}, a_{ 2}, a_{ 3}, ...,a_{ n}, ...
If we refer to example 1, a_{ 1} = 2, a_{ 2} = 4, a_{ 3} = 6 and so on. a_{ n} is the n^{ th} term.
A sequence may also be thought of a function whose domain is the set of positive integers and may therefore be defined by a formula that gives the n^{ th} term.
Example 3: The first 4 terms of a sequence defined by a_{ n} = 2n + (1)^{ n} are:
a_{ 1} = 2(1) + (1)^{ 1} = 1
a_{ 2} = 2(2) + (1)^{ 2} = 5
a_{ 3} = 2(3) + (1)^{ 3} = 5
a_{ 4} = 2(4) + (1)^{ 4} = 9
A sequence may also be defined recursively.
Example 4: Find the first 5 terms of a sequence defined recursively as follows:
a_{ 1} = 1, a_{ 2} = 3, a_{ n} = 2 a_{ n1}  a_{ n2} for n > 2
a_{ 1} = 1 , given
a_{ 2} = 3 , given
a_{ 3} = 2 a_{ 31}  a_{ 32} = 2 a_{ 2}  a_{ 1} = 2(3)  1 = 5
a_{ 4} = 2 a_{ 41}  a_{ 42} = 2 a_{ 3}  a_{ 2} = 2(5)  3 = 7
a_{ 5} = 2 a_{ 51}  a_{ 52} = 2 a_{ 4}  a_{ 3} = 2(7)  5 = 9
BSummation Notation.
Let a_{ 1}, a_{ 2}, a_{ 3}, ..., a_{ n} be the terms of a finite sequence. A convenient notation for the sum of the terms of the sequence uses the sigma notation since it uses the Greek letter sigma, written as ∑.
Example 5: The sum of the n terms in the sequence a_{ 1}, a_{ 2}, a_{ 3}, ..., a_{ n} is written as
n 
∑ a_{i} = a_{1} + a_{2} + a_{3} + ... + a_{ n} 
i = 1 
Example 6: Find the sum S of the first 7 terms in the sequence 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...
S = 2+4+6+8+10+12+14 = 56
Example7: Find the sum S of the first 5 terms in the sequence whose term a_{ n} is defined by a_{ n} = (2n+1)/n.
 5 
S=  ∑ (2n+1)/n = (2*1+1)/1 + (2*2+1)/2 + (2*3+1)/3 + (2*4+1)/4 + (2*5+1)/5 = 737 / 60
  i = 1 
Exercises:
1. What are the first 4 terms of a sequence defined by a_{n} = n^{2}  n?
2. Find the first 6 terms of a sequence defined recursively as follows:
a_{ 1} = 1, a_{ 2} = 3, a_{ n} =  a_{ n1}  2 a_{ n2} for n > 2
3. Find the sum S of the first 5 terms in the sequence whose term a_{ n} is defined by a_{ n} = n^{2} + (n + 1)/2.
Answers to above exercises
1. 0, 2, 6, 12
2. 1, 2, 4, 0, 8, 8
3. 65
