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Sum basis Binomial Theorem in One variable for Positive exponent


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


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.


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.


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.

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

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


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

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