Chapter 7 — Binomial Theorem

The Binomial Theorem provides a formula to expand (a + b)ⁿ for any positive integer n without repeated multiplication. It states (a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + ... + ⁿCₙbⁿ, where ⁿCᵣ are binomial coefficients.

Chapter 7 introduces the Binomial Theorem, which enables efficient expansion of binomials (a + b)ⁿ for any positive integer n. The chapter explains how the coefficients follow Pascal's triangle pattern and can be calculated using combinations (ⁿCᵣ). It covers the theorem's proof via mathematical induction, special cases like (x – y)ⁿ and (1 + x)ⁿ, and practical applications including evaluating large powers like (98)⁵ and proving divisibility properties. The coefficients are binomial coefficients with the property that there are always (n+1) terms in an expansion, with powers of the first term decreasing and the second increasing.

1. 01Binomial Theorem: (a + b)ⁿ = ΣⁿCₖaⁿ⁻ᵏbᵏ from k=0 to n, where ⁿCᵣ = n!/(r!(n–r)!)
2. 02Pascal's triangle and binomial coefficients provide expansion coefficients without multiplication
3. 03There are always (n+1) terms in (a + b)ⁿ expansion, with indices of a and b summing to n in each term
4. 04Special cases: (x – y)ⁿ uses alternating signs; (1 + x)ⁿ simplifies to ⁿC₀ + ⁿC₁x + ⁿC₂x² + ...
5. 05Practical applications include computing large powers (e.g., (98)⁵ = 9039207968) and proving divisibility (e.g., 6ⁿ – 5ⁿ ≡ 1 mod 25)
6. 06The theorem simplifies via the summation notation and is proven using mathematical induction