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Sequences and Series

Introduction

Sequences and series are fundamental topics in precalculus, providing a foundation for calculus and mathematical analysis. A sequence is an ordered list of numbers, while a series is the sum of the terms of a sequence.

Sequences

A sequence is a function whose domain is the set of positive integers. It can be defined explicitly by a formula or recursively by relating each term to previous terms.

1. Arithmetic Sequences

An arithmetic sequence is a sequence in which the difference between consecutive terms is constant.
  • General formula:
    where:
    • is the th term,
    • is the first term,
    • is the common difference,
    • is the term number.

  • Example: If and , then:

2. Geometric Sequences

A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a constant ratio .
  • General formula:
    where:
    • is the th term,
    • is the first term,
    • is the common ratio,
    • is the term number.

  • Example: If and , then:

3. Recursive Sequences

Some sequences are defined using previous terms rather than a formula.
  • Example: Fibonacci Sequence:

Factorial and Summation Notation

Factorial notation is used to represent the product of all positive integers up to a given number :
As a special case, zero factorial is defined as .

  • Example:
Summation notation (sigma notation) represents the sum of terms in a sequence:
where is the index of summation, is the upper limit of summation, and is the lower limit of summation.

  • Example:

Series

A series is the sum of the terms in a sequence.

1. Arithmetic Series

The sum of the first terms of an arithmetic sequence is given by:
Or equivalently:
  • Example: Find the sum of the first terms of the sequence :

2. Geometric Series

The sum of the first terms of a geometric sequence is:
  • Example: Find the sum of the first 5 terms of the sequence :

3. Infinite Geometric Series

If , the sum of an infinite geometric series is:
  • Example: Find the sum of :

Applications of Sequences and Series

  1. Finance – Calculating compound interest using geometric sequences.
  2. Physics – Modeling population growth and radioactive decay.
  3. Engineering – Signal processing and wave patterns.
  4. Computer Science – Algorithm analysis and recursion.
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