Binomial Theorem - Notes, Concept and All Important Formula

BINOMIAL THEOREM

$(x+y)^{n}={ }^{n} C_{0} x^{n}+{ }^{n} C_{1} x^{n-1} y+{ }^{n} C_{2} x^{n-2} y^{2}+\ldots$$. .+{ }^{n} C_{r} x^{n-r} y^{r}+\ldots$$. .+{ }^{n} C_{n} y^{n}$ $=\displaystyle \sum_{ r =0}^{ n }{ }^{ n } C _{ r } x ^{ n - r } y ^{ r }$, where $n \in N$.

1. IMPORTANT TERMS IN THE BINOMIAL EXPANSION ARE :

(a) General term: The general term or the $(r+1)^{\text {th }}$ term in the expansion of $(x+y)^{n}$ is given by

$T _{ r +1}={ }^{ n } C _{ r } x ^{ n \cdot r } \cdot y ^{r}$

(b) Middle term :

The middle term (s) is the expansion of $(x+y)^{n}$ is (are) :

(i) If $n$ is even, there is only one middle term which is given by $T _{( n +2) / 2}={ }^{ n } C _{ n / 2} \cdot x ^{ n / 2} \cdot y ^{ n / 2}$

(ii) If $n$ is odd, there are two middle terms which are $T _{( n +1) / 2}$ & $T _{[( n +1) / 2]+1}$

(c) Term independent of x :

Term independent of $x$ contains no $x$; Hence find the value of r for which the exponent of $x$ is zero.

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All Chapter Notes, Concept and Important Formula

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2. SOME RESULTS ON BINOMIAL COEFFICIENTS :

(a) ${ }^{n} C_{x}={ }^{n} C_{y} \Rightarrow x=y$ or $x+y=n$

(b) ${ }^{ n } C _{ r -1}+{ }^{ n } C _{ r }={ }^{ n +1} C _{ r }$

(c) $C _{0}+ C _{1}+ C _{2}+\ldots \ldots= C _{ n }=2^{ n }, C _{ r }={ }^{ n } C _{ r }$

(d) $C _{0}+ C _{2}+ C _{4}+\ldots$$\ldots= C _{1}+ C _{3}+ C _{5}+\ldots$$\ldots=2^{ n -1}, C _{ r }$ $={ }^{ n } C _{ r }$

(e) $C _{0}^{2}+ C _{1}^{2}+ C _{2}^{2}+\ldots$$\ldots+ C _{ n }^{2}$$={ }^{2 n} C _{ n }$$=\dfrac{(2 n ) !}{ n ! n !}, C _{ r }$$={ }^{ n } C _{ r }$

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3. Greatest coefficient & greatest term in expansion of $(x+a)^{n}$ :

(a) If $n$ is even, greatest binomial coefficient is ${ }^{n} C_{n / 2}$

If $n$ is odd, greatest binomial coefficient is ${ }^{ n } C _{\frac{ n -1}{2}}$ or ${ }^{ n } C _{\frac{ n +1}{2}}$

(b) For greatest term :

Greatest Term $= \begin{cases}T_p \,\&\, T_{p+1} & \text{if $ \frac{n+1}{\left|\frac{x}{a}\right|+1}$ is an integer equal to p}\\T_{q+1} & \text{if $ \frac{n+1}{\left| \frac{x}{a}\right|+1}$is a non integer & $\in (q,q+1), $} \\& \text{$q\in I$} \end{cases}$

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4. BINOMIAL THEOREM FOR NEGATIVE OR FRACTIONAL INDICES:

If $n \in R$, then $(1+x)^{n}$ $=1+n x+\dfrac{n(n-1)}{2 !} x^{2}+\dfrac{n(n-1)(n-2)}{3 !} x^{3}+\ldots$ $\infty$ provided $|x|<1$

Note :

(i) $(1-x)^{-1}=1+x+x^{2}+x^{3}+\ldots \ldots \ldots \ldots \infty$

(ii) $(1+x)^{-1}=1-x+x^{2}-x^{3}+\ldots \ldots \ldots \ldots \infty$

(iii) $(1-x)^{-2}=1+2 x+3 x^{2}+4 x^{3}+\ldots \ldots \ldots \ldots \infty$

(iv) $(1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\ldots \ldots \ldots \ldots \infty$

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5. EXPONENTIAL SERIES :

(a) $e^{x}=1+\dfrac{x}{1 !}+\dfrac{x^{2}}{2 !}+\dfrac{x^{3}}{3 !}+\ldots \ldots . \infty ;$ where $x$ may be any real or complex number & $e=\displaystyle \lim_{n \rightarrow \infty}\left(1+\dfrac{1}{n}\right)^{n}$

(b) $a ^{ x }=1+\dfrac{ x }{1 !} \ln a +\dfrac{ x ^{2}}{2 !} \ln ^{2} a +\dfrac{ x ^{3}}{3 !} \ln ^{3} a +\ldots \ldots \infty$, where $a >0$

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6. LOGARITHMIC SERIES :

(a) $\ln (1+x)=x-\dfrac{x^{2}}{2}+\dfrac{x^{3}}{3}-\dfrac{x^{4}}{4}+\ldots \ldots \infty$, where $-1<x \leq 1$

(b) $\ln (1- x )=- x -\dfrac{ x ^{2}}{2}-\dfrac{ x ^{3}}{3}-\dfrac{ x ^{4}}{4}-\ldots \ldots . \infty$, where $-1 \leq x <1$

(c) $\ln \dfrac{(1+ x )}{(1- x )}=2\left( x +\dfrac{ x ^{3}}{3}+\dfrac{ x ^{5}}{5}+\ldots \ldots . \infty\right),| x |<1$

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