Arithmetic Sequence Summation Methods

Overview

This lecture covers methods to find the sum of an arithmetic sequence using standard formulas, with step-by-step examples and explanations of each formula’s application.

Sum Formulas for Arithmetic Sequences

• The sum of the first n terms (Sₙ) of an arithmetic sequence can be found by Sₙ = n/2 × (a₁ + aₙ).
• Alternatively, use Sₙ = n/2 × [2a₁ + (n − 1)d] when the nth term (aₙ) is not given.
• a₁ is the first term, aₙ is the nth (last) term, n is the total number of terms, and d is the common difference.

Example Calculations

• If a₁ = 5, a₁₀ = 68, n = 10: S₁₀ = 10/2 × (5 + 68) = 5 × 73 = 365.
• If a₁ = 5, d = –7, n = 10: S₁₀ = 10/2 × [2×5 + (10–1)×(–7)] = 5 × (10 + (–63)) = 5 × (–53) = –265.
• For sequence 3, 5, 7, ... (a₁ = 3, d = 2), n = 10: S₁₀ = 10/2 × [2×3 + 9×2] = 5 × 24 = 120.
• For a₁ = 6, d = –3, n = 10: S₁₀ = 10/2 × [2×6 + 9×(–3)] = 5 × (12 + (–27)) = 5 × (–15) = –75.

Summation Notation Examples

• For Σ6n, n from 1 to 31: a₁ = 6, a₃₁ = 186, n = 31, Sₙ = 31/2 × (6 + 186) = 31 × 96 = 2976.
• For Σ(3n–1), n from 3 to 20: a₁ = 8, a₁₈ = 59, n = 18, S₁₈ = 9 × (8 + 59) = 9 × 67 = 603.
• For Σ(3n+4), n from 2 to 16: a₁ = 10, a₁₅ = 52, n = 15, S₁₅ = 15/2 × (10 + 52) = 15 × 31 = 465.
• For Σ(n–5), n from 2 to 24: a₁ = –3, a₂₃ = 19, n = 23, S₂₃ = 23/2 × (–3 + 19) = 23 × 8 = 184.

Key Terms & Definitions

• Arithmetic Sequence — A sequence where the difference between consecutive terms is constant.
• Sₙ (Sum) — The sum of the first n terms of a sequence.
• a₁ (First Term) — The initial term of the sequence.
• aₙ (Nth Term) — The last term in the sum (or the nth term).
• d (Common Difference) — The fixed amount between terms.
• n (Number of Terms) — The count of terms included in the sum.

Action Items / Next Steps

• Practice similar problems using both sum formulas.
• Review how to identify a₁, d, n, and aₙ from sequence notation or formula.
• Prepare for upcoming exercises on arithmetic series.