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General and Middle Terms

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General and Middle Terms - Lesson Summary

Binomial Theorem:

(x+y)n = nC0 xn + nC1 xn-1y + nC2 xn-2y2 + ......... + nCn-1 xyn-1 + nCn yn

First term = nC0 xn

Second term = nC1 xn-1 y

Third term = nC2 xn-2 y2

Seventh term = nC6 xn-6 y6

(r+1)th term = T r+1 = nCr xn-r yr

This is known as the general term of the expansion which can be used to find any term in the expansion.

Middle term(s) in the binomial expansion

If the number of terms in the expansion is odd then there will be one middle term.

If the number of terms in the expansion is even, then there will be two middle terms.

The number of middle terms in an expansion depends upon the index of the binomial.

Number of terms in (x + y)n is (n+1).

Case I: If n is even, then the number of terms in the expansion is odd.

Middle term = (n/2 + 1)th term

Ex: If the index is 6, then the (6/2 + 1)th = 4th term is the middle term.

Case II: If n is odd, then the number of terms in the expansion is even.

Middle terms = ((n+1)/2)th and (((n+1)/2) + 1)th terms

Ex:

If the index is 5, then the middle terms are ((5+1)/2)th = 3rd term and (((5+1)/2) + 1)th = 4th term.

Constant term

A term without any variable is known as a term independent of x or a constant term.

Ex:

(x + 1/x)2 = x2 + 2.x.1/x + (1/x)2

= x2 + 2 + (1/x)2

The 2nd term is the constant term.

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