Tutorial 11: Binomial Theorem

Tutorial problems covering binomial expansion and applications.

Sections

Binomial Expansions (Problems 1-4)

  • Expanding $(a + b)^n$ for positive integers
  • Using Pascal's triangle
  • Computing binomial coefficients

Finding Specific Terms (Problems 5-8)

  • General term formula: $T_{r+1} = \binom{n}{r}a^{n-r}b^r$
  • Finding particular coefficients
  • Middle terms and specific positions

General Binomial Theorem (Problems 9-12)

  • Expansion for negative/fractional indices
  • $(1 + x)^n$ for $|x| < 1$
  • Approximation applications

Key Formulas

Standard Binomial: $$(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r}b^r$$

General Term: $$T_{r+1} = \binom{n}{r} a^{n-r}b^r$$

Binomial Coefficient: $$\binom{n}{r} = \frac{n!}{r!(n-r)!}$$

General Binomial (for any real $n$): $$(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \cdots$$

Links