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If \[{{(\sqrt{A}+B)}^{n}}=I+f\] where \[l\] and \[n\] are positive integers, \[n\] being odd and \[A-{{B}^{2}}=K>0\] then \[(I+f)\,.\,f={{K}^{n}}\]   where \[A-{{B}^{2}}=K>0\] and \[\sqrt{A}-B<1\].  

• If \[n\] is even integer then  \[{{(\sqrt{A}+B)}^{n}}+{{(\sqrt{A}-B)}^{n}}=I+f+{f}'\]

  Hence L.H.S. and I are integers. \[\therefore \] \[f+{f}'\] is also integer   \[\Rightarrow \]\[f+{f}'=1\]; \[\therefore \]\[{f}'=(1-f)\]   Hence \[\left| \,\frac{y}{x}\, \right|<1\] \[={{(\sqrt{A}+B)}^{n}}{{(\sqrt{A}-B)}^{n}}\]   \[={{(A-{{B}^{2}})}^{n}}={{K}^{n}}\].

(1) Use of differentiation : This method applied only when the numericals occur as the product of binomial coefficients.   Solution process :  (i) If last term of the series leaving the plus or minus sign be m, then divide m by \[n\] if \[q\] be the quotient and \[r\] be the remainder. i.e., \[m=nq+r\].   Then replace \[x\] by \[{{x}^{q}}\] in the given series and multiplying both sides of expansion by \[{{x}^{r}}\].   (ii) After process (i), differentiate both sides, w.r.t. \[x\] and put \[x=1\] or \[-1\] or \[i\] or \[-i\] etc. according to given series.   (iii) If product of two numericals (or square of numericals) or three numericals (or cube of numericals) then differentiate twice or thrice.   (2) Use of integration : This method is applied only when the numericals occur as the denominator of the binomial coefficients.   Solution process : If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x\]\[+{{C}_{2}}{{x}^{2}}+\] \[.....+{{C}_{n}}{{x}^{n}}\] , then we integrate both sides between the suitable limits which gives the required series.   (i) If the sum contains \[{{C}_{0}},\,{{C}_{1}},\,{{C}_{2}},.......\,{{C}_{n}}\] with all positive signs, then integrate between limit 0 to 1.   (ii) If the sum contains alternate signs (i.e. +, –) then integrate between limit \[-1\] to 0.   (iii) If the sum contains odd coefficients i.e., \[({{C}_{0}},{{C}_{2}},{{C}_{4}}.....)\] then integrate between \[-1\] to 1.   (iv) If the sum contains even coefficients (i.e., \[{{C}_{1}},\,{{C}_{3}},\,{{C}_{5}}.....)\] then subtracting (ii) from (i) and then dividing by 2.   (v) If denominator of binomial coefficients is product of two numericals then integrate two times, first taking limit between 0 to \[x\] and second time take suitable limits.

In the binomial expansion of \[{{(1+x)}^{n}},\]   \[\,{{(1+x)}^{n}}{{=}^{n}}{{C}_{0}}{{+}^{n}}{{C}_{1}}x{{+}^{n}}{{C}_{2}}{{x}^{2}}+.....+{{\,}^{n}}{{C}_{r}}{{x}^{r}}+....+{{\,}^{n}}{{C}_{n}}{{x}^{n}}\]   where \[^{n}{{C}_{0}},{{\,}^{n}}{{C}_{1}},{{\,}^{n}}{{C}_{2}},......,{{\,}^{n}}{{C}_{n}}\] are the coefficients of various powers of \[x\] and called binomial coefficients, and they are written as \[{{C}_{0}},\,{{C}_{1}},\,{{C}_{2}},\,.....{{C}_{n}}\].   Hence, \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+.....\]   \[+{{C}_{r}}{{x}^{r}}+.....+{{C}_{n}}{{x}^{n}}\].....(i)   (1) The sum of binomial coefficients in the expansion of \[{{(1+x)}^{n}}\] is \[{{2}^{n}}\].   Putting \[x=1\] in (i), we get \[{{2}^{n}}={{C}_{0}}+{{C}_{1}}+{{C}_{2}}+.....+{{C}_{n}}\]      .....(ii)   (2) Sum of binomial coefficients with alternate signs : Putting \[x=-1\] in (i)   We get, \[{{C}_{0}}-{{C}_{1}}+{{C}_{2}}-{{C}_{3}}+......=0\]                        …..(iii)   (3) Sum of the coefficients of the odd terms in the expansion of \[{{(1+x)}^{n}}\] is equal to sum of the coefficients of even terms and each is equal to \[{{2}^{n-1}}\].   (4) \[^{n}{{C}_{r}}=\frac{n}{r}{{\,}^{n-1}}{{C}_{r-1}}=\frac{n}{r}\,.\frac{n-1}{r-1}{{\,}^{n-2}}{{C}_{r-2}}\] and so on.   (5) Sum of product of coefficients in the expansion is \[^{2n}{{C}_{n+r}}\].   Hence,\[{{C}_{0}}{{C}_{r}}+{{C}_{1}}{{C}_{r+1}}+......{{C}_{n-r}}.{{C}_{n}}{{=}^{2n}}{{C}_{n+r}}\]= \[\frac{2n!}{(n-r)!\text{ }(n+r)!}\]                .....(iv)   (6) Sum of squares of coefficients : Putting \[{{(a+b)}^{m}}={{a}^{m}}+m{{a}^{m-1}}b\] in (iv), we get  \[^{2n}{{C}_{n}}=C_{0}^{2}+C_{1}^{2}+......+C_{n}^{2}\]   (7) \[^{n}{{C}_{r}}{{+}^{n}}{{C}_{r-1}}{{=}^{n+1}}{{C}_{r}}\]   (8) \[{{C}_{1}}+2{{C}_{2}}+3{{C}_{3}}+...........+\,n.\,{{C}_{n}}=n\,.\,{{2}^{n-1}}\]   (9) \[{{C}_{1}}-2{{C}_{2}}+3{{C}_{3}}-................=0\]   (10) \[{{C}_{0}}+2{{C}_{1}}+3{{C}_{2}}+........+(n+1){{C}_{n}}=(n+2){{2}^{n-1}}\]   (11) \[C_{0}^{2}+C_{1}^{2}+C_{2}^{2}+...........+C_{n}^{2}=\frac{(2n)\,!}{{{(n\,!)}^{2}}}\]   (12) \[C_{0}^{2}-C_{1}^{2}+C_{2}^{2}-C_{3}^{2}+...........=\left\{ \begin{matrix} 0,\,\,\text{if }n\text{ is odd}\,\,\,\,\,\,\,\,\,\,\,  \\ {{(-1)}^{n/2}}{{.}^{n}}{{C}_{n/2}},\,\text{if }n\,\text{is even}  \\ \end{matrix} \right.\]

(1) Greatest term : If \[{{T}_{r}}\] and \[{{T}_{r+1}}\] be the \[{{r}^{th}}\] and \[{{(r+1)}^{th}}\] terms in the expansion of \[{{(1+x)}^{n}}\], then   \[\frac{{{T}_{r+1}}}{{{T}_{r}}}=\frac{^{n}{{C}_{r}}{{x}^{r}}}{^{n}{{C}_{r-1}}{{x}^{r-1}}}=\frac{n-r+1}{r}x\]   Let numerically, \[{{T}_{r+1}}\] be the greatest term in the above expansion. Then \[{{T}_{r+1}}\ge {{T}_{r}}\] or \[\frac{{{T}_{r+1}}}{{{T}_{r}}}\ge 1\].   \[\therefore \]  \[\frac{n-r+1}{r}\,|x|\ge 1\]   or  \[r\le \frac{(n+1)}{(1+|x|)}\,|x|\]        …..(i)   Now substituting values of \[n\] and \[x\] in (i), we get \[r\le m+f\] or  \[r\le m\] , where \[m\] is a positive integer and \[f\] is a fraction such that \[0<f<1\].   When n is even \[{{T}_{m+1}}\] is the greatest term, when \[n\] is odd \[{{T}_{m}}\] and \[{{T}_{m+1}}\] are the greatest terms and both are equal.   Short cut method : To find the greatest term (numerically) in the expansion of \[{{(1+x)}^{n}}\].   (i) Calculate \[m=\left| \,\frac{x(n+1)}{x+1}\, \right|\]   (ii) If m is integer, then \[\frac{2r!}{{{(r!)}^{2}}}\] and \[{{T}_{m+1}}\] are equal and both are greatest term.   (iii) If m is not integer, then \[{{T}_{[m]+1}}\] is the greatest term, where [.] denotes the greatest integral part.   (2) Greatest coefficient   (i) If \[n\] is even, then greatest coefficient is \[^{n}{{C}_{n/2}}\]   (ii) If \[n\] is odd, then greatest coefficient are \[^{n}{{C}_{\frac{n+1}{2}}}\] and \[^{n}{{C}_{\frac{n+3}{2}}}\].

In the expansion of \[{{\left( {{x}^{\alpha }}\pm \frac{1}{{{x}^{\beta }}} \right)}^{n}}\], if \[{{x}^{m}}\] occurs in \[{{T}_{r+1}}\], then \[r\] is given by \[n\alpha -r(\alpha +\beta )=m\] \[\Rightarrow \] \[r=\frac{n\alpha -m}{\alpha +\beta }\]   Thus in above expansion if constant term which is independent of \[x,\] occurs in \[\frac{2n!}{(n-r)!\text{ }(n+r)!}\] then \[r\] is determined by   \[n\alpha -r(\alpha +\beta )=0\]\[\Rightarrow \]\[r=\frac{n\alpha }{\alpha +\beta }\]

The middle term depends upon the value of n.   (1) When n is even, then total number of terms in the expansion of \[{{(x+y)}^{n}}\] is \[n+1\] (odd). So there is only one middle term i.e., \[{{\left( \frac{n}{2}+1 \right)}^{\text{th}}}\] term is the middle term. \[{{T}_{\left[ \frac{n}{2}+1 \right]}}{{=}^{n}}{{C}_{n/2}}{{x}^{n/2}}{{y}^{n/2}}\]   (2) When n is odd, then total number of terms in the expansion of \[{{(x+y)}^{n}}\] is \[n+1\] (even). So, there are two middle terms i.e.,\[{{\left( \frac{n+1}{2} \right)}^{\text{th}}}\] and \[{{\left( \frac{n+3}{2} \right)}^{\text{th}}}\] are two middle terms.    \[{{T}_{\left( \frac{n+1}{2} \right)}}{{=}^{n}}{{C}_{\frac{n-1}{2}}}{{x}^{\frac{n+1}{2}}}{{y}^{\frac{n-1}{2}}}\]  and \[{{T}_{\left( \frac{n+3}{2} \right)}}{{=}^{n}}{{C}_{\frac{n+1}{2}}}{{x}^{\frac{n-1}{2}}}{{y}^{\frac{n+1}{2}}}\]  

• When there are two middle terms in the expansion then their binomial coefficients are equal.

 

• Binomial coefficient of middle term is the greatest binomial coefficient.

 

\[{{(a+b+c)}^{n}}\]can be expanded as : \[{{C}_{0}}-{{C}_{1}}+{{C}_{2}}-{{C}_{3}}+......=0\]   \[={{(a+b)}^{n}}{{+}^{n}}{{C}_{1}}{{(a+b)}^{n-1}}{{(c)}^{1}}{{+}^{n}}{{C}_{2}}{{(a+b)}^{n-2}}{{(c)}^{2}}+.....+{{\,}^{n}}{{C}_{n}}\,{{c}^{n}}\]   \[=(n+1)\,\text{term }+n\,\text{term }+\text{ }(n-1)\text{term }+...+1\text{term}\]   \[\therefore \] Total number of terms = \[(n+1)+(n)+(n-1)+......+1=\frac{(n+1)(n+2)}{2}\].   Similarly, number of terms in the expansion of   \[{{(a+b+c+d)}^{n}}=\frac{(n+1)(n+2)(n+3)}{6}\].  

Independent term or constant term of a binomial expansion is the term in which exponent of the variable is zero.   Condition : \[(n-r)\] [Power of \[x]\,+\] [Power of \[y]=0,\] in the expansion of \[{{[x+y]}^{n}}\].  

The general term of the expansion is \[{{(r+1)}^{th}}\] term usually denoted by \[{{T}_{r+1}}\] and \[{{T}_{r+1}}{{=}^{n}}{{C}_{r}}{{x}^{n-r}}{{y}^{r}}\]  

• In the binomial expansion of \[{{(x-y)}^{n}},\,{{T}_{r+1}}={{(-1)}^{r}}{{\,}^{n}}{{C}_{r}}{{x}^{n-r}}{{y}^{r}}\]

 

• In the binomial expansion of \[{{(1+x)}^{n}},\,{{T}_{r+1}}{{=}^{n}}{{C}_{r}}{{x}^{r}}\]

 

• In the binomial expansion of \[{{(1-x)}^{n}},\,{{T}_{r+1}}={{(-1)}^{r}}{{\,}^{n}}{{C}_{r}}{{x}^{r}}\]

 

• In the binomial expansion of \[{{(x+y)}^{n}}\], the pth term from the end is \[{{(n-p+2)}^{th}}\] term from beginning.

 

(1) Replacing\[y\]y  by \[-y\]   in (i), we get,   \[{{(x-y)}^{n}}={{\,}^{n}}{{C}_{0}}{{x}^{n-0}}{{y}^{0}}{{-}^{n}}{{C}_{1}}{{x}^{n-1}}{{y}^{1}}{{+}^{n}}{{C}_{2}}{{x}^{n-2}}{{y}^{2}}-.......\]\[+{{(-1)}^{r}}{{\,}^{n}}{{C}_{r}}{{x}^{n-r}}{{y}^{r}}+....+{{(-1)}^{n}}{{\,}^{n}}{{C}_{n}}{{x}^{0}}{{y}^{n}}\]   i.e., \[{{(x-y)}^{n}}=\sum\limits_{r=0}^{n}{{{(-1)}^{r}}{{\,}^{n}}{{C}_{r}}{{x}^{n-r}}{{y}^{r}}}\]                              The terms in the expansion of \[{{(x-y)}^{n}}\] are alternatively positive and negative, the last term is positive or negative according as \[n\] is even or odd.   (2) Replacing \[x\] by 1 and y by \[x\] in equation (i) we get,  \[{{(1+x)}^{n}}{{=}^{n}}{{C}_{0}}{{x}^{0}}{{+}^{n}}{{C}_{1}}{{x}^{1}}{{+}^{n}}{{C}_{2}}{{x}^{2}}+......+{{\,}^{n}}{{C}_{r}}{{x}^{r}}+......{{+}^{n}}{{C}_{n}}{{x}^{n}}\]   i.e., \[{{(1+x)}^{n}}=\sum\limits_{r=0}^{n}{^{n}{{C}_{r}}{{x}^{r}}}\]   This is expansion of \[{{(1+x)}^{n}}\] in ascending power of \[x\].   (3) Replacing \[x\] by 1 and \[y\] by \[-x\] in (i) we get,    \[{{(1-x)}^{n}}={{\,}^{n}}{{C}_{0}}{{x}^{0}}-{{\,}^{n}}{{C}_{1}}{{x}^{1}}+{{\,}^{n}}{{C}_{2}}{{x}^{2}}-.....\]  \[+{{(-1)}^{r}}{{\,}^{n}}{{C}_{r}}{{x}^{r}}+....+{{(-1)}^{n}}{{\,}^{n}}{{C}_{n}}{{x}^{n}}\]    i.e.,  \[{{(1-x)}^{n}}=\sum\limits_{r=0}^{n}{{{(-1)}^{r}}{{\,}^{n}}{{C}_{r}}{{x}^{r}}}\]   (4) \[{{(x+y)}^{n}}+{{(x-y)}^{n}}=\]\[2\,{{[}^{n}}{{C}_{0}}{{x}^{n}}{{y}^{0}}{{+}^{n}}{{C}_{2}}{{x}^{n-2}}{{y}^{2}}\]\[{{+}^{n}}{{C}_{4}}{{x}^{n-4}}{{y}^{4}}+.......]\] and   \[{{(x+y)}^{n}}-{{(x-y)}^{n}}=2\,{{[}^{n}}{{C}_{1}}{{x}^{n-1}}{{y}^{1}}\]\[{{+}^{n}}{{C}_{3}}{{x}^{n-3}}{{y}^{3}}+{{\,}^{n}}{{C}_{5}}{{x}^{n-5}}{{y}^{5}}+...]\]   (5) The coefficient of \[{{(r+1)}^{th}}\] term in the expansion of \[{{(1+x)}^{n}}\] is \[^{n}{{C}_{r}}\].   (6) The coefficient of \[{{x}^{r}}\] in the expansion of \[{{(1+x)}^{n}}\] is \[^{n}{{C}_{r}}\].