Sum basis Binomial Theorem in One variable for Positive exponent

Formula

$(1+x)^n$ $\,=\,$ $1$ $+$ $nx$ $+$ $\dfrac{n(n-1)}{2!}x^2$ $+$ $\dfrac{n(n-1)(n-2)}{3!}x^3$ $+$ $\cdots$

Introduction

Let $x$ be a variable and $n$ be a constant. They both belongs to real numbers group, which means $x ∈ R$ and $n ∈ R$. Adding one to variable $x$ forms a binomial $1+x$. The constant $n$ belongs to real numbers $n ∈ R$. The binomial $1+x$ raised to the power $n$ forms a power function in the following form.

$(1+x)^n$

This power function is called the sum basis binomial theorem in one variable for positive exponent. It can be expanded mathematically in terms of variable $x$ as follows.

Expansion

The sum basis binomial theorem in one variable is expanded as follows.

$(1+x)^n$ $\,=\,$ $1$ $+$ $nx$ $+$ $\dfrac{n(n-1)}{2!}x^2$ $+$ $\dfrac{n(n-1)(n-2)}{3!}x^3$ $+$ $\cdots$

The above expansion is also written in terms of binomial coefficients as follows. You can follow any one of the following or different notation for binomial coefficient in the expansion.

$(1).\,\,\,$ $(1+x)^n$ $\,=\,$ $\displaystyle \binom{n}{0}$ $+$ $\displaystyle \binom{n}{1} x$ $+$ $\displaystyle \binom{n}{2} x^2$ $+$ $\displaystyle \binom{n}{3} x^3$ $+$ $\cdots$ $+$ $\displaystyle \binom{n}{n} x^n$

$(2).\,\,\,$ $(1+x)^n$ $\,=\,$ $^nC_0$ $+$ $^nC_1 x$ $+$ $^nC_2 x^2$ $+$ $^nC_3 x^3$ $+$ $\cdots$ $+$ $^nC_n x^n$

Simple form

Instead, the expansion of sum basis binomial theorem in one variable is easily written as follows.

$(1+x)^n$ $\,=\,$ $\displaystyle \sum_{r \,=\, 0}^{n} \dfrac{n(n-1)(n-2)\cdots (n-r+1)}{r!}x^r$

Similarly, the expansion of this binomial theorem can also be written in the following popular forms.

$(1).\,\,\,$ $(1+x)^n$ $\,=\,$ $\displaystyle \sum_{r \,=\, 0}^{n} \binom{n}{r} x^r$

$(2).\,\,\,$ $(1+x)^n$ $\,=\,$ $\displaystyle \sum_{r \,=\, 0}^{n}$ $^nC_r\,x^r$

Proof

Learn how to derive the sum basis binomial theorem in one variable for positive index in mathematics.