JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Greatest Term and Greatest Coefficient

(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 \[^{n}{{C}_{\frac{n+3}{2}}}\].