Arithmetic Series Lecture Notes

Introduction

• Objective: Find the sum of an arithmetic series using a formula.
• Example series: 4, 7, 10, 13, 16, 19, 22... to 58.

Formula for Sum of Arithmetic Series

• Formula: ( S_n = \frac{a_1 + a_n}{2} \times n )
  • (a_1): First term
  • (a_n): Last term
  • (n): Number of terms

Finding the Number of Terms (n)

• Use: ( a_n = a_1 + (n-1) \times d )
• Given:
  • (a_1 = 4)
  • (a_n = 58)
  • Common difference, d = 3
• Calculation:
  • (58 = 4 + (n-1) \times 3)
  • Simplify: (58 = 3n + 1)
  • Solve for (n):
    • (57 = 3n)
    • (n = 19)

Calculating the Sum

• Apply to formula: ( S_n = \frac{4 + 58}{2} \times 19 )
• ( S_n = 31 \times 19 = 589 )

Example 2: Decreasing Series

• Series: 288, ..., 16 (decreasing by 4)
• Calculation:
  • (a_1 = 288)
  • (a_n = 16)
  • d = -4
  • Find (n):
    • (16 = 288 + (n-1) \times -4)
    • (16 = -4n + 292)
    • (n = 69)
• Sum:
  • (S_n = \frac{288 + 16}{2} \times 69 = 152 \times 69 = 10,488)

Practice Problem

• Series: 96, 89, 82, 75, ..., 12
• Calculation:
  • (a_1 = 96)
  • (a_n = 12)
  • d = -7
  • Find (n):
    • (12 = 96 + (n-1) \times -7)
    • (n = 13)
• Sum:
  • (S_n = \frac{96 + 12}{2} \times 13 = 54 \times 13 = 702)

Conclusion

• Steps to find the sum:
  1. Use the formula for (n).
  2. Apply the sum formula.
• Additional resources and video content available on video-t.net

These notes cover how to find the sum of arithmetic series using formulas to determine the number of terms and calculate the sum.