MathematicsChapter 7: Mathematical Induction and Binomial Theorem

BINOMIAL THEOREM (POSITIVE INTEGER n)

(a+b)^n = sum_{r=0}^{n} nCr a^{n-r} b^r .

Learning objectives

State the main ideas and notation used in binomial theorem (positive integer n).
Apply the rules of binomial theorem (positive integer n) in a clear, step-by-step solution.
Check signs, conditions, and the final answer for mathematical accuracy.

Definition

(a+b)ⁿ = sum_r=0ⁿ nCr a^n-r b^r . There are n+1 terms. Coefficients are a row of Pascal's triangle. Meaning: you expand by choosing b from r of the n brackets (the rest contribute a). a and b can be numbers or letters, including (1+x)^n and (2x-3y)^n.

Mathematical form

a=x, b=2, want x^4-r * 2^r with 4-r=2 so r=2

T = 4C2 x² 2²

n = sum

r=2 so r

Where / symbols

BINOMIAL THEOREM (POSITIVE INTEGER n): (a+b)^n = sum_{r=0}^{n} nCr a^{n-r} b^r . There are n+1 terms. Coefficients are a row of Pascal's triangle. Meaning: you expand by choosing b from r of the n brackets (the rest contribute a). a and b can be numbers or letters, including (1+x)^n and (2x-3y)^n.
1. THE SUM: r runs 0 to n. First term a^n, last term b^n. General term T_{r+1} = nCr a^{n-r} b^r (the (r+1)th term, r starting at 0).
2. SMALL CHECK: (a+b)^2 = a^2+2ab+b^2. (a+b)^3 = a^3+3a^2 b+3 a b^2+b^3. Pascal: 1,4,6,4,1 for n=4.

Overview

Write the sum, then a small n:

The Sum

r runs 0 to n. First term aⁿ, last term bⁿ. General term T_r+1 = nCr a^n-r b^r (the (r+1)th term, r starting at 0).

Examples T_{r+1} general · nC0=1 first · nCn=1 last

Small Check

(a+b)² = a²+2ab+b². (a+b)³ = a³+3a² b+3 a b²+b³. Pascal: 1,4,6,4,1 for n=4.

Examples n=3: 1 3 3 1 · Signs if (a-b): (-1)^r · (a-b)^n = sum nCr a^{n-r} (-b)^r

WHAT — BINOMIAL THEOREM (POSITIVE INTEGER n)

RULE — 1. THE SUM

r runs 0 to n. First term aⁿ, last term bⁿ. General term T_r+1 = nCr a^n-r b^r (the (r+1)th term, r starting at 0).

METHOD — 2. SMALL CHECK

(a+b)² = a²+2ab+b². (a+b)³ = a³+3a² b+3 a b²+b³. Pascal: 1,4,6,4,1 for n=4.

WORKED EXAMPLE

Worked: (x+2)⁴, term in x². a=x, b=2, want x^4-r 2^r with 4-r=2 so r=2 T = 4C2 x² 2² = 6 x² * 4 = 24 x^2

How to answer a binomial theorem (positive integer n) question

Identify the given quantities, notation, and required result.
Choose the matching rule and write it before substitution.
Work one algebraic step at a time, keeping brackets and signs clear.
Check restrictions, simplify the result, and verify it where possible.

Worked example

(x+2)^4, term in x^2.
a=x, b=2, want x^4-r * 2^r with 4-r=2 so r=2
T = 4C2 x² 2²
= 6 * x² * 4 = 24 x^2

In simple words

The power of a plus the power of b is always n. Use that to find r.

Key points

n+1 terms
coeff of a^n-r b^r is nCr
r = 0 to n
Pascal row n

Exam preparation

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