Series (Copy)
12 Series – Cheat Sheet
12.1 Binomial Theorem
- Expansion of (a + b)ⁿ, where n is a positive integer:
(a + b)ⁿ = Σ (nCr × aⁿ⁻ʳ × bʳ) , where r = 0 to n
- Coefficient simplification required.
12.2 General Term
- Tᵣ₊₁ = nCr × aⁿ⁻ʳ × bʳ, 0 ≤ r ≤ n
- Example: Find the term independent of x in (x + 1/x²)¹⁰.
- General term: Tᵣ₊₁ = ¹⁰Cr × x¹⁰⁻ʳ × (x⁻²)ʳ = ¹⁰Cr × x¹⁰⁻ʳ⁻²ʳ = ¹⁰Cr × x¹⁰⁻³ʳ
- For term independent of x: 10 − 3r = 0 → r = 10/3 (if integer, substitute to find coefficient).
12.3 Arithmetic Progression (AP)
- nth term: uₙ = a + (n − 1)d
- Sum of first n terms: Sₙ = n/2 × [2a + (n − 1)d]
OR Sₙ = n/2 × (first term + last term)
12.4 Geometric Progression (GP)
- nth term: uₙ = a × rⁿ⁻¹
- Sum of first n terms: Sₙ = a(1 − rⁿ) / (1 − r), r ≠ 1
12.5 Sum to Infinity of GP
- Condition: |r| < 1
- Formula: S∞ = a / (1 − r)