## Binomial Theorem

# General and Middle Terms

- The general term in the expansion of (x + a)
^{n}is^{n}C_{r}. x^{n-r}. a^{r} - The number of terms in the expansion of (a + b + c + d)
^{n}is \tt \frac{\left(n+1\right)\left(n+2\right)\left(n+3\right)}{3!} - The number of terms in (a
_{1}+ a_{2}+ a_{3}+ - - - - + a_{r})^{n}is^{n+r-1}C_{r-1} - If ‘n’ is odd there will be ‘2’ middle terms in the expansion (x + a)
^{n}which are \tt \left(\frac{n+1}{2}\right)^{th}\ and \left(\frac{n+3}{2}\right)^{th} - If ‘n’ is even there will be one middle term in the expansion (x + a)
^{n}which is \tt \left(\frac{n}{2}+1\right)^{th} term - The binomial coefficient of the middle term is the greatest binomial coefficient in the expansion of (x + a)
^{n} - The term independent of ‘x’ in the expansion of \tt \left[ax^{P}+\frac{b}{x^{q}}\right]^{n} is T
_{r+1}where \tt r=\frac{np}{p+q} (integer) - The term containing the coefficient of x
^{m}in the expansion of \tt \left(ax^{p}+\frac{b}{x^{q}}\right)^{n} is T_{r+1}where \tt r=\frac{np-m}{p+q} (integer) - r
^{th}term from the end in the expansion of (x+y)^{n}is (n−r+2)^{th}term from beginning. - Numerically greatest term in the expansion of (1 + x)
^{n}where ‘n’ is a positive integer if \tt \frac{\left(n+1\right)\mid x \mid}{1+\mid x\mid}=P then P^{th}term and (P + 1)^{th}term are the two numerically greatest terms in (1 + x)^{n}. Also lt_{P}l = lt_{P+1}l - Suppose in (1 + x)
^{n}expansion if \tt \frac{\left(n+1\right)\mid x \mid}{1+\mid x\mid}=P+F where P is an integer and 0 < F < 1, then (P+1)^{th}term is the numerically greatest term in (1 + x)^{n}.

### Part1: View the Topic in this video From 08:53 To 12:00

### Part2: View the Topic in this video From 00:10 To 09:27

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1. General term in the expansion of (x + a)^{n} is given by

T^{r+1} = ^{n}C_{r} x^{n−r} a^{r}.

2. If n is even, then in the expansion of (x + a)^{n}, the middle term is \left(\frac{n}{2}+1\right)th term.

3. If n is odd, then in the expansion of (x + a)^{n}, the middle terms are \frac{\left(n+1\right)}{2}th term and \frac{\left(n+3\right)}{2}th term.

4. The term independent of x in the expansion of \left(ax^{p}+\frac{b}{x^{q}}\right)^{n} is T_{r+1}, where r=\frac{np}{p+q}

5. In the expansion of (x + a)^{n}

\frac{T_{r+1}}{T_{r}}=\frac{n-r+1}{r}\times \frac{a}{x}