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BINOMIAL THEOREM

Algebraic Identities

THE BINOMIAL THEOREM

If x, y aer two complex numbers then the binomial theorem for any positive integer index n is given by

 

(x + y) n = xn + nC1 xn-1y + nC2xn-2y2 + . . . + nCnyn

PROPERTIES OF BINOMIAL THEOREM

1. In a binomial expansion for any integer n there are always n+1 terms.
2. In any term of a binomial expansion the sum of the exponents of x and y is always equal to n.
3. The term ⁿC₂x^n-2y² is the (r+1^th) term from the beginning of the series. This means that the third term of a the series (x+y)n is given by ⁿC₂x^n-2y².

The (r+1)^th term in the above expansion is

Tr+1 = nCr xn-ryr

4. The binomial coefficient of terms equidistant from the beginning and the end are equal

nCr = nCn-r (where 0<=r<=n)

5. The greatest term is given by

TG = [(n+1)y/(x+y)]

6. In the expansion of (x+y)n if n is even then there is only on middle term and if n is odd then we have two middle terms ie, the (n+1)/2^th term and (n+3)/2^th term.

If n is even then TMid = nCn/2xn/2 yn/2

If n is odd then TMid(1) = nC(n-1)/2x(n-1)/2 y(n-1)/2 If n is odd then TMid(2) = nC(n-3)/2x(n-3)/2 y(n-3)/2

Factoring Formulae

which extends to the general formulae

BINOMIAL EXPANSION FOR A NEGATIVE COMPLEX NUMBER Y

(x - y)n = nC0xn - nC1xn-1y + nC2xn-2y2. . . + nCn(-1)nyn

The (r+1)^th term in this expansion is

Tm+1 = (-1)m.nCm xn-m ym

SOME STANDARD RESULTS

The expansion of (1 + x)ⁿ is

(1+x)n = C0 + C1x + C2x2 + . . . . . . Cnxn

Where C_m = ⁿC_m

(1-x)n = C0 - C1x + C2x2 - C3x3 + . . . + (-1)nCnxn

Putting x = 1 in each of the above expansions, we get

C0 + C1 + C2+ . . .+ Cn-1 + Cn = 2n

C0 - C1 + C2- . . . . + (- 1)nCn = 0

On adding these two we get

C0 + C2+ C4 + . . . = 2n-1

On subtracting we get

C1 + C3 + C5 + . . . = 2n-1

All of the above results are used in deriving other identities involving binomial coefficients.