Sunday, December 14, 2008

Some Deductions from Binomial Theorem

1. total number of terms in the expansion = n+1

2. The sum of indices of x and a in each term is n.

3. the coefficients of terms equidistant from the beginning and the end are equal.

4. (x-a)n
= (r = 0 to n)Σ ((-1) r*nCrxn-rar

The terms in the expansion of(x-a)n are alternatively positive and negative, the last term is positive or negative according as n is even or odd.

5. (1+x) n = (r = 0 to n)Σ nCrxr

you get it by putting x =1 and a = x in the expression for (x+a)n.

6. (x+1) n = (r = 0 to n)Σ nCrxn-r

7. (1-x) n = (r = 0 to n)Σ(-1)r* nCrxr

8. (x+a) n +(x-a) n = 2[nC0xna0 +
nC2xn-2a2 +
nC4xn-4a4+ ...]

9. General term in a binomial expansion

(r+1) term in (x+a) n
= nCrxn-rar

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