 # General and Middle Terms

• The general term in the expansion of (x + a)n is nCr. xn-r . ar
• 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 (a1 + a2 + a3 + - - - - + ar)n is n+r-1Cr-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 Tr+1 where \tt r=\frac{np}{p+q} (integer)
• The term containing the coefficient of xm in the expansion of \tt \left(ax^{p}+\frac{b}{x^{q}}\right)^{n} is Tr+1 where \tt r=\frac{np-m}{p+q} (integer)
• rth 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 Pth term and (P + 1)th term are the two numerically greatest terms in (1 + x)n. Also ltPl = ltP+1l
• 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.

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1. General term in the expansion of (x + a)n is given by
Tr+1 = nCr xn−r ar.

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 Tr+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}