Binomial theorem — A/L Combined Mathematics

What must I remember about Binomial theorem?

• Binomial Theorem for any rational index n: If n is rational and |x| < 1, then (1 + x)ⁿ = 1 + nx + [n(n-1)/2!]x² + [n(n-1)(n-2)/3!]x³ + ... to infinity.
• Convergence condition: The infinite binomial series converges if and only if |x| < 1 (-1 < x < 1); if x does not satisfy this, the expression must be rearranged to factor out the dominant term.
• General term for rational index: The (r + 1)-th term is T_r+1 = [n(n-1)(n-2)...(n - r + 1) / r!] * x^r.
• Key special series: (1 - x)^-1 = 1 + x + x² + x³ + ...; (1 + x)^-1 = 1 - x + x² - x³ + ...; (1 - x)^-2 = 1 + 2x + 3x² + 4x³ + ...; (1 + x)^-2 = 1 - 2x + 3x² - 4x³ + ...
• Approximations using Binomial expansion: For small x (|x| << 1), truncate after linear or quadratic terms: (1 + x)ⁿ ≈ 1 + nx or 1 + nx + (n(n-1)/2)x², used for calculating square roots and percentages.

Test yourself (1 minute)

1. Factoring for valid domain

   To expand (a + bx)ⁿ where |bx/a| < 1, factor out aⁿ: (a + bx)ⁿ = aⁿ * [1 + (b/a)x]ⁿ; expanding directly without factoring yields divergent nonsense.

2. Calculating roots to decimal precision

   To approximate √2, write √2 = √(49/25 * 50/49) = (7/5)√(1 + 1/49) = 1.4 * (1 + 1/49)^1/2, where x = 1/49 is very small.

3. Finite vs Infinite series distinction

   When n is a positive integer, the series terminates after (n + 1) terms; when n is negative or fractional, the series has infinitely many terms and requires |x| < 1.

Practise Binomial theorem questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 22

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Where does Binomial theorem fit in the syllabus?

Year 2 · Term 1