Sequences and Series - The Binomial Theorem (AS Unit 1: Pure Mathematics A) — WJEC A-Level Mathematics
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Sequences and Series - The Binomial Theorem (AS Unit 1: Pure Mathematics A) explained
For a positive integer n, the Binomial Theorem expands (a + bx)ⁿ into a finite series of n + 1 terms: (a + bx)ⁿ = aⁿ + (n choose 1) a^(n−1)(bx) + (n choose 2) a^(n−2)(bx)² plus higher order terms up to (bx)ⁿ.
Read the full explanation
The general term is (n choose r) a^(n−r)(bx)ʳ for r = 0, 1, ..., n. Evaluating terms requires enclosing the entire term (bx) in brackets so that both coefficient and variable are raised to the power: (bx)ʳ = bʳ xʳ. The expansion is exact and valid for all real values of x because the index n is a non-negative integer. Applications include expanding products such as (1 + 2x)(2 − x)⁵ up to a specified power of x by multiplying selected terms, and calculating approximations to decimal powers like (1.02)¹⁰ by substituting small values of x.
Your focus
- Expand (a + bx)ⁿ for positive integers n up to a specified power of x.
- Calculate individual coefficients in a binomial expansion without expanding fully.
- Use binomial expansions to approximate numerical powers by substituting small values of x.
Sequences and Series - The Binomial Theorem (AS Unit 1: Pure Mathematics A) exam tips
Marking Points
- selecting and applying the correct binomial coefficients (n choose r)
- raising both the coefficient b and variable x to the required power r in (bx)ʳ
- evaluating coefficients accurately and expressing the expansion in ascending powers of x
- multiplying selected terms when expanding products of polynomials with binomials
Examiner Tips
- 💡Always put brackets around the second term: write (bx)ʳ before evaluating numerical powers.
- 💡Write down the first four terms carefully, simplifying each coefficient before combining.
Common Mistakes
- forgetting to square or cube the numerical coefficient of x, writing 2x² instead of (2x)² = 4x²
- making sign errors when expanding terms with a negative sign inside the bracket, such as (−3x)³