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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.


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


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


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


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 \(a >0\)


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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