Sequences and Summations

The following are some notes taken during a lecture.

Sequences: Definition

Sequences are ordered lists of elements.
- 1, 2, 3, 5, 8
- 1, 3, 9, 27, 81, …….
Sequences arise throughout mathematics, computer science, and in many other disciplines, ranging from botany to music.
We introduce the terminology to represent sequences and sums of the terms in the sequences.
Definition: A sequence is a function from a subset of the integers (usually either the set {0, 1, 2, 3, 4, …..} or {1, 2, 3, 4, ….} ) to a set S.
The notation an is used to denote the image of the integer n. We can think of an as the equivalent of f(n) where f is a function from {0,1,2,…..} to S. We call an a term of the sequence.

Example:

Geometric Progression

Definition: A geometric progression is a sequence of the form: where the initial term a and the common ratio r are real numbers.

Examples:
Let a = 1 and r = −1. Then:
Let a = 2 and r = 5. Then:
Let a = 6 and r = 1/3. Then:

Arithmetic Progression

Definition: A arithmetic progression is a sequence of the form: where the initial term a and the common difference d are real numbers.

Examples:

Let a = −1 and d = 4:
Let a = 7 and d = −3:
Let a = 1 and d = 2:

Strings

Definition: A string is a finite sequence of characters from a finite set (an alphabet).

Sequences of characters or bits are important in computer science.

The empty string is represented by λ.

The string abcde has length 5.

Recurrence Relation

Definition: A recurrence relation for the sequence {an} is an equation that expresses an in terms of one or more of the previous terms of the sequence, namely, a0, a1, …, an-1, for all integers n with n ≥ n0, where n0 is a nonnegative integer.

A sequence is called a solution of a recurrence relation if its terms satisfy the recurrence relation.
The initial conditions for a sequence specify the terms that precede the first term where the recurrence relation takes effect.

Example 1: Let {an} be a sequence that satisfies the recurrence relation an = an-1 + 3 for n = 1,2,3,4,…. and suppose that a0 = 2. What are a1 , a2 and a3?

[Here a0 = 2 is the initial condition.]

Solution: We see from the recurrence relation that

a1 = a0 + 3 = 2 + 3 = 5
a2 = 5 + 3 = 8
a3 = 8 + 3 = 11

Fibronacci Sequence

Definition: Define the Fibonacci sequence, f0 ,f1 ,f2,…, by:

Initial Conditions: f0 = 0, f1 = 1

Recurrence Relation: fn = fn-1 + fn-2

Example: Find f2 ,f3 ,f4 , f5 and f6 .

Answer:

         f2 = f1 + f0 = 1 + 0 = 1,
          f3 = f2 + f1 = 1 + 1 = 2,
          f4 = f3 + f2 = 2 + 1 = 3,
          f5 = f4 + f3 = 3 + 2 = 5,
          f6 = f5 + f4 = 5 + 3 = 8.

Solving Recurrence Relations

Finding a formula for the nth term of the sequence generated by a recurrence relation is called solving the recurrence relation.
Such a formula is called a closed formula.

Method 1: Working upward, forward substitution

Let {an} be a sequence that satisfies the recurrence relation an = an-1 + 3 for n = 2,3,4,…. and suppose that a1 = 2.
      a2   = 2 + 3
      a3   = (2 + 3) + 3 = 2 + 3 ∙ 2
      a4   = (2 + 2 ∙ 3) + 3 = 2 + 3 ∙ 3
                    .
                    .
                    .
            an = an-1 + 3 = (2 + 3 ∙ (n – 2)) + 3 = 2 + 3(n – 1)

Sequences: Some Examples

Example: Suppose that a person deposits $10,000.00 in a savings account at a bank yielding 11% per year with interest compounded annually. How much will be in the account after 30 years?

Let Pn denote the amount in the account after 30 years. Pn satisfies the following recurrence relation:

Pn = Pn-1 + 0.11Pn-1 = (1.11) Pn-1
with the initial condition P0 = 10,000

Given a few terms of a sequence, try to identify the sequence. Conjecture a formula, recurrence relation, or some other rule.
Some questions to ask?
- Are there repeated terms of the same value?
- Can you obtain a term from the previous term by adding an amount or multiplying by an amount?
- Can you obtain a term by combining the previous terms in some way?
- Are they cycles among the terms?
- Do the terms match those of a well known sequence?

Special Integer Sequences

Example 1: Find formulae for the sequences with the following first five terms: 1, ½, ¼, 1/8, 1/16

Solution: Note that the denominators are powers of 2. The sequence with an = 1/2n is a possible match. This is a geometric progression with a = 1 and r = ½.

Example 2: Consider 1,3,5,7,9

Solution: Note that each term is obtained by adding 2 to the previous term. A possible formula is an = 2n + 1. This is an arithmetic progression with a =1 and d = 2.

Example 3: 1, -1, 1, -1,1

Solution: The terms alternate between 1 and -1. A possible sequence is an = (−1)n . This is a geometric progression with a = 1 and r = −1.

Common Sequences

Inferring Sequences

Example: Conjecture a simple formula for an if the first 10 terms of the sequence {an} are 1, 7, 25, 79, 241, 727, 2185, 6559, 19681, 59047.

Solution: Note the ratio of each term to the previous approximates 3. So now compare with the sequence 3n . We notice that the nth term is 2 less than the corresponding power of 3. So a good conjecture is that an = 3n − 2.

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