MathematicsChapter 7: Mathematical Induction and Binomial Theorem

BINOMIAL FOR NEGATIVE OR FRACTION n (SCHOOL)

For |x|<1, (1+x)^n = 1 + n x + n(n-1)/2!

Learning objectives

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

Definition

For |x|<1, (1+x)ⁿ = 1 + n x + n(n-1)/2! x² + n(n-1)n-23! x^3 + ... even if n is not a positive integer. You usually keep the first few terms (approximation). (1-x)^{-1} = 1+x+x^2+... . Meaning: the same pattern of falling factors, but it does not stop. Only valid as an infinite series in a disc |x|<1 in school wording.

Mathematical form

n=-1, 1 + (-1)x + (-1)-22 x² + ...

n = 1 + n x + n(n-1)/2

n=-1

Where / symbols

BINOMIAL FOR NEGATIVE OR FRACTION n (SCHOOL): For |x|<1, (1+x)^n = 1 + n x + n(n-1)/2! x^2 + n(n-1)(n-2)/3! x^3 + ... even if n is not a positive integer. You usually keep the first few terms (approximation). (1-x)^{-1} = 1+x+x^2+... . Meaning: the same pattern of falling factors, but it does not stop. Only valid as an infinite series in a disc |x|<1 in school wording.
1. FIRST TERMS: (1+x)^{1/2} ~ 1 + (1/2)x - (1/8)x^2 + ... . Compute n(n-1)/2 for the x^2 coeff. Stop at the term the question asks.
2. USES: Approximate sqrt(1.02) as (1+0.02)^{1/2} with two terms. (1-x)^{-1} geometric. Do not plug |x|>1 into the infinite expansion as if it were exact.

Overview

Write three terms, then a named series:

First Terms

(1+x)12} ~ 1 + (12)x - (18)x^2 + ... . Compute n(n-1)/2 for the x^2 coeff. Stop at the term the question asks.

Examples n=-1: all +1 coeffs if (1-x)^{-1} · n=1/2 square root · Show 2! 3! in the bottoms

Uses

Approximate sqrt(1.02) as (1+0.02)12} with two terms. (1-x)^-1 geometric. Do not plug |x|>1 into the infinite expansion as if it were exact.

Examples sqrt(1+x) approx 1+x/2 · Need |x|<1 · Finite n still uses the finite theorem

WHAT — BINOMIAL FOR NEGATIVE OR FRACTION n (SCHOOL)

RULE — 1. FIRST TERMS

(1+x)12} ~ 1 + (12)x - (18)x^2 + ... . Compute n(n-1)/2 for the x^2 coeff. Stop at the term the question asks.

METHOD — 2. USES

Approximate sqrt(1.02) as (1+0.02)12} with two terms. (1-x)^-1 geometric. Do not plug |x|>1 into the infinite expansion as if it were exact.

WORKED EXAMPLE

Worked: first three terms of (1+x)^-1. n=-1, 1 + (-1)x + (-1)-22 x^2 + ... = 1 - x + x^2 - ... matches 11+x geometric with ratio -x if |x|<1

How to answer a binomial for negative or fraction n (school) 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

first three terms of (1+x)^-1.
n=-1, 1 + (-1)x + (-1)-22 x² + ...
= 1 - x + x^2 - ...
matches 11+x geometric with ratio -x if |x|<1

In simple words

Infinite binomial needs |x|<1 in this course. Positive integer n does not need that.

Key points

falling product / r!
|x|<1 for the infinite story
stop where asked
small x

Exam preparation

Exam Focus

Quick revision

Revision Zone

Review the key facts, formulas and checks before you move on.