Math 21 Lecture Notes on Arithmetic Series

Introduction

• Speaker: Sergio Venezuela Jr.
• Focus: Arithmetic Series

Learning Goals

1. Differentiate between arithmetic sequence and arithmetic series.
2. Define and illustrate an arithmetic series.
3. Identify each element of the formula for arithmetic series.
4. Solve problems involving arithmetic series.

Formula for Arithmetic Series

• Formulas: Two formulas for arithmetic series are introduced.

First Formula

• Formula:
  [ S_n = \frac{n}{2} \times (2a_1 + (n - 1)d) ]

  • Where:
    • ( S_n ) = sum of the series
    • ( n ) = number of terms
    • ( a_1 ) = first term
    • ( d ) = common difference

• Explanation:

  • The sum of the series is equal to half the number of terms multiplied by the sum of twice the first term and the product of the common difference and one less than the number of terms.

Second Formula

• Formula:
  [ S_n = \frac{n}{2} \times (a_1 + a_n) ]

  • Where:
    • ( a_n ) = last term of the series

• Explanation:

  • The sum of the entire series is equal to half the number of terms multiplied by the sum of the first and last term.

Problem Solving Example 1

• Problem Statement: Find the sum of the first 10 terms where the series is 4, 10, 16,...

• Given:

  • First term (a_1) = 4
  • Common difference (d) = 6
  • Number of terms (n) = 10

• Using the First Formula:

  • Plug in values:
    [ S_{10} = \frac{10}{2} \times (2 \cdot 4 + (10 - 1) \cdot 6) ]
  • Simplification:
  • ( S_{10} = 5 \times (8 + 54) = 5 \times 62 = 310 )

• Conclusion: The sum of the first 10 terms is 310.

Problem Solving Example 2

• Problem Statement: Find how many terms are needed for the series 20, 18, 16,... to sum to -100.

• Given:

  • First term (a_1) = 20
  • Common difference (d) = -2 (since series is decreasing)
  • Sum of series (S) = -100

• Using the First Formula:

  • Plugging into the sum formula:
    [ -100 = \frac{n}{2} \times (40 - 2(n - 1)) ]
  • Rearrange and solve for n:
  • Arrived at the equation:
    [ 2n^2 - 21n - 100 = 0 ]

• Factoring:

  • Factors found: ( (n + 4)(n - 25) = 0 )
  • Solutions: ( n = -4 ) (not valid) and ( n = 25 )

• Conclusion: There are 25 terms in the series required to achieve a sum of -100.

Problem Solving Example 3

• Problem Statement: Find the sum of odd integers from 1 to 99.

• Given:

  • First term (a_1) = 1
  • Last term (a_n) = 99
  • Number of terms (n) = 50

• Using the Second Formula:

  • Plugging into the sum formula: [ S_{50} = \frac{50}{2} \times (1 + 99) = 25 \times 100 = 2500 ]

• Conclusion: The sum of odd integers from 1 to 99 is 2500.

Key Takeaways

• Distinction between arithmetic sequence and series is crucial.
• Two key formulas for determining the sum of arithmetic series.
• Problem-solving strategies involve identifying given values and correctly applying the formulas.