Binomial theorem — A/L Combined Mathematics

Practical applications of the Binomial Theorem encompass finding specific terms, determining the term independent of x, identifying greatest coefficients, and calculating numerical approximations. The general term formula T_r+1 = ⁿC_r a^n-r b^r allows direct extraction of the coefficient of x^k by setting the net exponent of x to k. To find the term independent of x, the power of x is equated to zero. To determine the numerically greatest term in the expansion of (1 + x)ⁿ, the ratio of consecutive terms |T_r+1 / T_r| = [(n - r + 1) / r] |x| is compared against 1. For small |x| ≪ 1, the expansion provides accurate linear and quadratic approximations.

Check before you answer (A/L Pure Maths Paper 1)

1. Power Extraction Equating: Collecting all powers of x into a single exponent expression and setting it to 0 for independent terms or k for specified powers.
2. Greatest Term Ratio Test: Solving |T_r+1 / T_r| ≥ 1 for integer r to establish the greatest term index.
3. Decimal Approximations: Expanding terms up to x³ and substituting small decimals (e.g., x = 0.01) to evaluate values like (1.01)¹⁰ to specified decimal accuracy.