Skip to main content

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


Labels:

Comments

Post a Comment

Popular posts from this blog

Inverse Trigonometric function - Notes, Concept and All Important Formula

INVERSE TRIGONOMETRIC FUNCTION 1. DOMAIN, RANGE & GRAPH OFINVERSE TRIGONOMETRIC FUNCTIONS : (a) \(\mathrm{f}^{-1}:[-1,1] \rightarrow[-\pi / 2, \pi / 2]\) , \(\mathrm{f}^{-1}(\mathrm{x})=\sin ^{-1}(\mathrm{x})\) (b) \(\mathrm{f}^{-1}:[-1,1] \rightarrow[0, \pi]\) , \(\mathrm{f}^{-1}(\mathrm{x})=\cos ^{-1} \mathrm{x}\) (c) \(\mathrm{f}^{-1}: \mathrm{R} \rightarrow(-\pi / 2, \pi / 2)\) , \(\mathrm{f}^{-1}(\mathrm{x})=\tan ^{-1} \mathrm{x}\) (d) \(\mathrm{f}^{-1}: \mathrm{R} \rightarrow(0, \pi)\) \(\mathrm{f}^{-1}(\mathrm{x})=\cot ^{-1} \mathrm{x}\) (e) \(\mathrm{f}^{-1}:(-\infty,-1] \cup[1, \infty)\) \(\rightarrow[0, \pi / 2) \cup(\pi / 2, \pi]\) , \(\mathrm{f}^{-1}(\mathrm{x})=\sec ^{-1} \mathrm{x}\) (f) \(\mathrm{f}^{-1}:(-\infty,-1] \cup[1, \infty)\) \(\rightarrow[-\pi / 2,0) \cup(0, \pi / 2]\) \(\mathrm{f}^{-1}(\mathrm{x})=\operatorname{cosec}^{-1} \mathrm{x}\) All Chapter Notes, Concept and Important Formula 2. PROPERTIES OF INVERSE CIRCULAR FUNCTIONS: Property-1 : (i) \(y=\sin ...
Post a Comment
Read more

Trigonometry Ratios and Identities - Notes, Concept and All Important Formula

TRIGONOMETRIC RATIOS & IDENTITIES Table Of Contents 1. RELATION BETWEEN SYSTEM OF MEASUREMENT OF ANGLES : \(\dfrac{D}{90}=\dfrac{G}{100}=\dfrac{2 C}{\pi}\) 1 Radian \(=\dfrac{180}{\pi}\) degree \(\approx 57^{\circ} 17^{\prime} 15^{\prime \prime}\) (approximately) 1 degree \(=\dfrac{\pi}{180}\) radian \(\approx 0.0175\) radian All Chapter Notes, Concept and Important Formula 2. BASIC TRIGONOMETRIC IDENTITIES : (a) \(\sin ^{2} \theta+\cos ^{2} \theta=1\) or \(\sin ^{2} \theta=1-\cos ^{2} \theta\) or \(\cos ^{2} \theta=1-\sin ^{2} \theta\) (b) \(\sec ^{2} \theta-\tan ^{2} \theta=1\) or \(\sec ^{2} \theta=1+\tan ^{2} \theta\) or \(\tan ^{2} \theta=\sec ^{2} \theta-1\) (c) If \(\sec \theta+\tan \theta\) \(=\mathrm{k} \Rightarrow \sec \theta-\tan \theta\) \(=\dfrac{1}{\mathrm{k}} \Rightarrow 2 \sec \theta\) \(=\mathrm{k}+\dfrac{1}{\mathrm{k}}\) (d) \(\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1\) or \(\operatorname{cosec}^{2} \theta=1+\cot ^{2} \th...
Post a Comment
Read more

Difference and relation between Differentiation and Integration

Relation between Differentiation and Integration Table Of Contents Look at the information given below. \[\mathbf{ y=f(x)}\] \[ \mathbf{ f'(x)\rightarrow \text{Derivatives of f(x)}}\] \[ \mathbf{\displaystyle \int_a^b f’(x) = ?}\] Can you tell me the value of above integral? Yes, it will be equal to f(b)-f(a) . We have already known this result. It tells us that integration is just the reverse of differentiation, integral of the derivative of the function f(x) is just equal to the difference in the function f(x) evaluated at the limits of integration. Indefinite integration- Notes and Formula Part 1 Now with this topic, we will understand how to apply this result to find the integral of a function.  Consider this function \(\mathbf{g(x)=x^2}\) Let's find the integral of this function from a to b i.e \(\mathbf{\displaystyle \int_a^b g(x) \, dx}\) .  Can you think how we can apply this result to find ...
Post a Comment
Read more