Arithmetic Series Lecture Notes

Definition

• An arithmetic series is the partial sum of a given number of terms in an arithmetic sequence.

Key Formulas

1. Sₙ = n * (a₁ + aₙ) / 2

   • Where:
     • Sₙ = sum of the arithmetic series
     • n = number of terms
     • a₁ = first term
     • aₙ = last term

2. Sₙ = n / 2 * (2a₁ + (n - 1) * d)

   • Where:
     • Sₙ = sum of the arithmetic series
     • n = number of terms
     • a₁ = first term
     • d = common difference

Formula Application

• Use the first formula when the first and last terms are known.
• Use the second formula when the first term and the common difference are known.

Example Problems

Problem 1: Sum of the First 20 Terms

• Given:
  • First term (a₁) = 5
  • Last term (a₂₀) = 62
• Using the first formula:
  • S₂₀ = 20 * (5 + 62) / 2
  • Simplifying:
    • S₂₀ = 20 * 67 / 2
    • S₂₀ = 10 * 67 = 670
• Result: The sum of the first 20 terms is 670.

Problem 2: Sum of the First 40 Terms

• Given sequence: 2, 5, 8, 11, ...
• Identify variables:
  • First term (a₁) = 2
  • Common difference (d) = 3 (5 - 2)
  • Number of terms (n) = 40
• Using the second formula:
  • S₄₀ = 40 / 2 * (2 * 2 + (40 - 1) * 3)
  • Simplifying:
    • S₄₀ = 20 * (4 + 117)
    • S₄₀ = 20 * 121
    • S₄₀ = 2420
• Result: The sum of the first 40 terms is 2420.

Conclusion

• Understanding and applying the formulas for arithmetic series is crucial for calculating sums efficiently.
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