Introduction to Sequences

Introduction to Sequences

Objectives

  • Define a sequence.
  • Determine the next term of a sequence.
  • Enumerate the terms in a sequence.

Activity: Complete the Incomplete

  • Identify the pattern and complete the missing terms.
  • Examples of identifying patterns:
    • OTTFFSSEN _? (T)
    • TOPHHON_? (D)
    • 1, 4, 9, 16, _ ? (25)
    • A, 1, B, 2, C, 3, _, _ (D, 4)
    • A B C D A _? (B)
    • A B C D C _? ()

Defining a Sequence

  • A sequence is a function whose domain is the finite set {1, 2, 3, 4, …, n } or the infinite set {1, 2, 3, 4 … } .
  • It may be generated from shapes, patterns, or rules.
  • It is a list of numbers having the first number, second number, third number, and so on.
  • Each item in a sequence is called a term.

Finite and Infinite Sequences

  • A sequence having a finite number of terms is called a finite sequence.
  • A sequence that has an infinite number of terms is called an infinite sequence.

Finding the Terms of a Sequence

  • Given the nth term, substitute the value of n for every n to find the terms of the sequence.

Example 1

  • Given: a_n = n + 2
  • Find the first five terms:
    • a_1 = 1 + 2 = 3
    • a_2 = 2 + 2 = 4
    • a_3 = 3 + 2 = 5
    • a_4 = 4 + 2 = 6
    • a_5 = 5 + 2 = 7
  • The first five terms are 3, 4, 5, 6, 7.

Example 2

  • Given: a_n = n^2 - 2n
  • Find the first five terms:
    • a_1 = 1^2 - 2(1) = -1
    • a_2 = 2^2 - 2(2) = 0
    • a_3 = 3^2 - 2(3) = 3
    • a_4 = 4^2 - 2(4) = 8
    • a_5 = 5^2 - 2(5) = 15
  • The first five terms are -1, 0, 3, 8, 15.

Let's Try This

  • Find the indicated term in the given sequence.
  • a_n = 2n + 1 (5th term)

Practice Exercises

  • Find the first five terms of a given sequence.
    1. a_n = 3 + (2n^2 - 1)
    2. a_n = 8n^2 + 4