Binomial Theorem for Positive Integral Indices

Binomial Theorem

7.1 Introduction

The Binomial Theorem provides a formula for expanding expressions of the form (a + b)ⁿ.

Historical Context: Known to ancient mathematicians, but fully developed by Isaac Newton.

Applications: Probability, algebra, calculus, and series expansions.

Simple Case:

(a + b)² = a² + 2ab + b²

(a + b)³ = a³ + 3a²b + 3ab² + b³

7.2 Binomial Theorem for Positive Integral Indices

For any positive integer n:

(a + b)ⁿ = ⁿC₀aⁿb⁰ + ⁿC₁a^n-1b¹ + ... + ⁿC_na⁰bⁿ

Or in sigma notation:

(a + b)ⁿ = Σ_k=0^{n n}C_k a^n-kb^k

Key Components:

• Binomial coefficients: ⁿC_k (also written as C(n,k) or (n choose k))
• Each term is of degree n
• Coefficients follow Pascal's Triangle pattern
• Number of terms is n+1

Pascal's Triangle (First 5 Rows):

        1
       1 1
      1 2 1
     1 3 3 1
    1 4 6 4 1
        

Each number is the sum of the two above it.

Term	General Form	Example: (x + y)4
1st	nC0anb0	1x4y0 = x4
2nd	nC1an-1b1	4x3y1 = 4x3y
3rd	nC2an-2b2	6x2y2
...	...	...
5th	nC4a0b4	1x0y4 = y4

Example 1: Expansion

Expand (2x + 3)³:

= ³C₀(2x)³3⁰ + ³C₁(2x)²3¹ + ³C₂(2x)¹3² + ³C₃(2x)⁰3³
= 1·8x³·1 + 3·4x²·3 + 3·2x·9 + 1·1·27
= 8x³ + 36x² + 54x + 27

Example 2: Finding Specific Term

Find the 4th term in the expansion of (x - 2y)⁵:

General term: T_r+1 = ⁵C_rx^5-r(-2y)^r
For 4th term (r=3): T₄ = ⁵C₃x²(-2y)³
= 10·x²·(-8y³) = -80x²y³

Properties of Binomial Coefficients:

• Sum of coefficients: 2ⁿ (set a = b = 1)
• Alternating sum: 0 (set a = 1, b = -1)
• Middle term:
  • If n is even: single middle term at k = n/2
  • If n is odd: two middle terms at k = (n-1)/2 and k = (n+1)/2

Example 3: Sum of Coefficients

Find the sum of coefficients in (2x - 3y)⁴:

Set x = y = 1:
Sum = (2·1 - 3·1)⁴ = (-1)⁴ = 1

Applications:

• Probability theory (binomial distribution)
• Approximations in calculus
• Combinatorics problems
• Financial mathematics (compound interest)