Sequence and Series: One-Page Formula Sheet

Sequence and Series — formula sheet

Formulas

• AP: aₙ = a + (n−1)d, Sₙ = n/2 [2a + (n−1)d]
• GP: Sₙ = a(rⁿ − 1)/(r − 1), S∞ = a/(1 − r)
• Σn = n(n+1)/2, Σn² = n(n+1)(2n+1)/6, Σn³ = [n(n+1)/2]²
• AM ≥ GM: (a + b)/2 ≥ √(ab)

Key ideas

• AP: constant difference; GP: constant ratio
• AM ≥ GM ≥ HM for positive numbers
• Sum to infinity of GP exists if |r| < 1
• Use method of differences for unusual series

Traps to avoid

• Using S∞ when |r| ≥ 1
• Off-by-one errors in n
• Applying AM–GM to negative numbers

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