Series and Sequences covers patterns in numbers — arithmetic and geometric progressions — and the powerful binomial expansion for raising expressions to powers.
Common difference d: uₙ = a + (n-1)d. Sum of n terms: Sₙ = n/2(2a + (n-1)d) = n/2(first + last). Applications: equally spaced values, linear growth. Find n-th term, sum, or number of terms given conditions.
Common ratio r: uₙ = arⁿ⁻¹. Sum of n terms: Sₙ = a(1-rⁿ)/(1-r). Sum to infinity (|r| < 1): S∞ = a/(1-r). Convergent when |r| < 1. Applications: compound interest, population growth, repeated percentage changes, bouncing balls.
(a+b)ⁿ = Σ ⁿCᵣ aⁿ⁻ʳ bʳ for positive integer n. ⁿCᵣ = n!/(r!(n-r)!). Useful for expanding (1+x)ⁿ or finding specific terms. (1+x)ⁿ for rational n (P3): 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + ... Valid for |x| < 1. Used for approximations: (1.02)¹⁰ ≈ 1 + 10(0.02) + ...
The sum to infinity S∞ = a/(1-r) is only valid for geometric progressions where |r| < 1 (i.e., -1 < r < 1). This means the terms get smaller and smaller, approaching zero, so the total converges to a finite value. If |r| ≥ 1, the terms do not decrease, and the sum grows without bound (diverges). Common exam question: "find the range of values of x for which a given GP converges" — set up |r| < 1 and solve. For the binomial expansion with rational index, the convergence condition is |x| < 1.
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