Unit 8: Sequences and Series

Objectives

• Define a sequence.

• Write an equation to determine a sequence.

• Define a series.

• Write summation notation to represent a series.

8.1 Defining Sequences and Series

Warm-Up

• Determine the pattern in each list of numbers below.
    - a. (not provided)
    - b. 1, 2.5, 4, 5.5, 7, …
    - c. (not provided)
    - d. 8, 5, 2, −1, −4, …
    - e. 1, 3, 9, 27, 81, …
    - f. 1, 4, 9, 16, 25, 36, …

Sequences

• Definition: A sequence is an ordered list of numbers that follow a certain pattern.
    - Examples from Warm-Up:
      - Sequence 1: 8, 5, 2, −1, −4, … (decreasing by 3 each time)
      - Sequence 2: 1, 4, 9, 16, 25, 36, … (perfect squares: $n^2$ for $n = 1, 2, 3, …$)

You Try

1. Determine the rule for each sequence:
      - a. 1, 1.5, 2, 2.5, 3, … (increasing by 0.5 each time)
      - b. −1, −8, −27, −64, … (cubic sequence: $n^3$ for $n = -1, -2, -3, …$)

2. Write the first six terms of the sequence that follows the rule given.
      - a. $n = 3 + 5$
        - First six terms: 3, 4, 5, 6, 7, 8
      - b. $n = (-2)^{-1}$
        - This seems to imply a misunderstanding; clarification may be needed.

Sequences in Context

• Example in a Video Game:
     - Start with 12 points; earn 2 more points for each minute active.
     - Rule for points based on the number of minutes active:
       - Let $m$ = number of minutes active.
       - Points after minutes = $12 + 2m$.
     - Graph this sequence on a Cartesian plane.

Series

• Definition: When the terms of a sequence are added together, the result is a series.
    - Types of Series:
      - Finite Series: a series with a limited number of terms.
        - Example: $3 + 5 + 7 + 9 + 11$
      - Infinite Series: a series that continues indefinitely.
        - Example: $3 + 5 + 7 + 9 + …$

Summation Notation

• Definition: A way to represent the sum of a series in a compact form.
    - Example: The series 3 + 5 + 7 + 9 + 11 can be written as $extstyle ext{Sum}<em>{i=1}^5 (2i + 1)$.   - Example: The infinite series can be noted as: $extstyle ext{Sum}</em>{n=1}^{ ext{∞}} (2n + 1)$.

Examples of Series in Summation Notation

1. Write each series using summation notation:
      - a. 25 + 50 + 75 + … + 250
        - $extstyle ext{Sum}<em>{i=1}^{n} (25i)$, where n is the number of terms.    - b. $rac{1}{2} + rac{2}{3} + rac{3}{4} + rac{4}{5} + …$      - General form: $extstyle ext{Sum}</em>{n=1}^{N} rac{n}{n+1}$.

You Try - Summation Notation

• Write each series using summation notation:
     - a. 5 + 15 + 25 + … + 135
       - Series can be represented as: $extstyle ext{Sum}<em>{i=1}^{n} (5i)$.    - b. 6 + 36 + 216 + 1296 + …      - Series can be represented as: $extstyle ext{Sum}</em>{i=1}^{n} (6^i)$.

Finding Sums of Series

1. Find the sum of the series using summation notation:
      - a. $extstyle ext{Sum}<em>{i=5}^{n} (3 + 2)$    [ \text{Sum} = (3+2) + … ]    - b. $extstyle ext{Sum}</em>{i=2}^{4} (4-1)$
      [ \text{Sum} = (4 - 1) + … ]

You Try - Problem Solving in Real Context

• You work in a grocery store, stacking apples in a square pyramid with the following configuration:
    - First layer: 1 apple
    - Second layer: 4 apples
    - Third layer: 9 apples
    - Continuing in this fashion, let the total number of apples be represented in summation notation:
      - $extstyle ext{Total} = ext{Sum}_{i=1}^{n} (i^2)$.

• To find the total number of apples in the pyramid, calculate:
     - Using a specific number of layers, say 7 layers: 1 + 4 + 9 + 16 + 25 + 36 + 49 = Total.

Assignment

• Complete Section 8.1 Practice A worksheet for further practice on sequences and series-related problems.