Lesson 5.1: Sequences and Sigma Notation

Introduction

In this lesson, we will explore the foundational concepts of sequences and sigma notation as part of our study of sequences, series, and financial mathematics. Our learning objectives today are:

• Understanding sequences defined by a formula and by a recurrence relation.
• Learning about sigma notation and standard summation results.
• Recognizing arithmetic and geometric sequences.
• Generating terms of a sequence from a formula or recurrence.
• Writing a sum in sigma notation and evaluating it.

So, let’s dive in!

What is a Sequence?

A sequence is an ordered list of numbers that follow a particular pattern. Each number in a sequence is called a term. Sequences can be defined in two main ways: through a formula or through a recurrence relation.

Sequences Defined by a Formula

A sequence defined by a formula means we have an explicit mathematical expression to find the $ n $-th term of the sequence. This type of sequence is often written in the form:

$$ a_n = f(n) $$

where $ a_n $ is the $ n $-th term and $ f(n) $ is an expression that describes how to calculate $ a_n $.

Example: Arithmetic Sequence

An arithmetic sequence is a sequence in which the difference between consecutive terms is constant. For example, consider the sequence defined by:

$$ a_n = 3 + (n - 1) \cdot 2 $$

This means that:

• For $ n = 1: a_1 = 3 + (1 - 1) \cdot 2 = 3 $
• For $ n = 2: a_2 = 3 + (2 - 1) \cdot 2 = 5 $
• For $ n = 3: a_3 = 3 + (3 - 1) \cdot 2 = 7 $

So, the first few terms are 3, 5, 7, 9, 11, ... and so on.

Sequences Defined by a Recurrence Relation

A sequence can also be defined by a recurrence relation, which expresses each term as a function of the preceding terms. The general form can be written as:

$$ a_n = g(a_{n-1}, a_{n-2}, \ldots) $$

where $ g $ is some function involving previous terms.

Example: Fibonacci Sequence

The Fibonacci sequence is a well-known example defined by the recurrence relation:

$$ a_n = a_{n-1} + a_{n-2}, \quad \text{for } n \geq 3 $$

with initial conditions:

$$ a_1 = 1, \quad a_2 = 1 $$

Calculating terms yields:

• $ a_3 = a_2 + a_1 = 1 + 1 = 2 $
• $ a_4 = a_3 + a_2 = 2 + 1 = 3 $
• $ a_5 = a_4 + a_3 = 3 + 2 = 5 $

Thus, the initial terms of the Fibonacci sequence are 1, 1, 2, 3, 5, 8, ...

Sigma Notation

Sigma notation is a concise way of representing summation, which is adding a sequence of numbers. The Greek letter sigma $ \Sigma $ denotes the sum.

Understanding Sigma Notation

The expression:

$$ \sum_{i=m}^{n} f(i) $$

means to sum the values of $ f(i) $ as $ i $ runs from $ m $ to $ n $. In this notation, $ m $ is the lower limit, and $ n $ is the upper limit of the summation.

Example: Summing an Arithmetic Series

Consider the sum of the first four terms of the sequence defined by:

$$ a_n = 3 + (n - 1) \cdot 2 $$

The terms are 3, 5, 7, and 9. We can express this sum in sigma notation as:

$$ S = \sum_{n=1}^{4} a_n = \sum_{n=1}^{4} (3 + (n - 1) \cdot 2) $$

Calculating gives:

egin{align*}

S & = a_1 + a_2 + a_3 + a_4 \

& = 3 + 5 + 7 + 9 \

& = $24\end{align*}$

Thus, $ S = 24 $.

Common Summation Results

Understanding standard summation results is crucial when working with sigmas, particularly applied to financial mathematics.

Example of Standard Sum Formulas

1. Sum of the First $ n $ Natural Numbers:

$$\sum_{i=1}^{n} i = \frac{n(n + 1)}{2}$$

1. Sum of the First $ n $ Squares:

$$\sum_{i=1}^{n} i^2 = \frac{n(n + 1)(2n + 1)}{6}$$

1. Sum of the First $ n $ Cubes:

$$\sum_{i=1}^{n} i^3 = \left( \frac{n(n + 1)}{2}

ight)^2$$

These formulas greatly simplify the computation of large sums.

Evaluating a Sum Using Summation Results

Let’s evaluate the sum:

$$ S = \sum_{i=1}^{10} i $$

Using the formula for natural numbers gives:

$$ S = \frac{10(10 + 1)}{2} = \frac{10 \cdot 11}{2} = 55 $$

Conclusion

In this lesson, we covered what sequences are, how they can be defined, and the power of sigma notation for simplifying summation processes. You learned how to generate specific terms of a sequence, articulate sums clearly using sigma notation, and apply standard summation results to facilitate calculations.

Study Notes

• A sequence is an ordered list of numbers.
• Sequences can be defined explicitly by formulas or recursively by relations.
• An arithmetic sequence has a constant difference between terms.
• Sigma notation allows us to represent sums concisely.
• Familiarity with standard summation results aids in evaluating sums efficiently.
• Practice generating sequences and using sigma notation for various applications, especially in financial contexts.