1. Binomial Theorem for positive integral index
If n is a positive integer and x, y is real or imaginary then (x+y)n = nC0 xn-0 y0 + nC1 xn-1 y1 + ..... +nCr xn-r yr + ......nCn x0 yr , i.e (x+y)n = ∑ nC0 xn-r yr
General term is Tr+1 = nCr xn-r yr Here nC0 , nC1 ,nCr , nCn are called the binomial coefficients.
Replacing y by -y the general term of (x-y)n is obtained as
Tr+1 = (-1)r nCr xn-r yr
Similarly, the general term of (1 + x)n and (1 - x)n can be obtained by replacing x by 1 and x by 1 and y by -x respectively.
nCr = nC(n-r)
nCr + nC(n-r) = n+1Cr
nCr / nCr-1 = (n - r + 1) / r
r (nCr) = n ( n-1Cr-1 )
nCr / (r+1) = ( n+1Cr+1 ) / (r+1)
2. Middle Term
The middle term depends upon the even or odd nature of n
Case 1: When n is even
Total number of terms in the expansion of (x+y)n is n+1 (odd)
So there is only 1 middle term i.e ((n/2) + 1)th term is the middle term
This is given by
T((n/2) + 1) = nC(n/2) x(n/2) y(n/2)
Case 2: When n is odd
Total number of terms in the expansion of (x+y)n is n+1(even)
So there are 2 middle terms i.e ((n + 1) / 2 )th and((n + 3) / 2 )thh terms are both middle terms
They are given by
T((n + 1) / 2) = nC((n-1) / 2) x((n+1) / 2) y((n-1) / 2)
T((n + 3) / 2) = nC((n+1) / 2) x((n-1) / 2) y((n+1) / 2)
3. Greatest Term
If Tr and Tr+1 be the rth and (r+1)th terms in the expansion, (x+y)n then
Tr+1 / Tr = (( n - r + 1 ) / r ) x
Let Tr+1 be the numerically greatest term in the above expression
(( n - r + 1 ) / r ) x
(a) If n is even, the greatest coefficient = nC(n/2)
(b) If n is odd, then greatest coefficients are and nC((n-1)/2) and nC((n+1)/2)
In the binomial expansion of (1 + x)n let us denote the coefficients by nC0 , nC1 ,nCr , nCn respectively
Since (1+x)n = nC0 + nC1 x + nC2 x2 + ..... +nCn xn
nC0 + nC1 + nC2 + ..... +nCn =2n
nC0 + nC2 + nC4 +.....=nC1 + nC3 + nC5
Sum of odd terms coefficients=Sum of even terms coefficients=2n-1
5. Series Summation
nC0 + nC1 + nC2 + ..... +nCn =2n
(b) nC0 - nC1 +nC2 - nC3 + ... + (-1)n nCn = 0
(c) nC0 + nC2 + nC4 +.....=nC1 + nC3 + nC5 = 2n-1
(d) nC20 + nC21 + nC22 + ..... +nC2n =(2n) ! /(n!)3
6. Multinomial Expansion
If n is a positive integer a1 , a2 ,... am are real or imaginary then
(a1 + a2 + a3 +....+am )n = ∑(n! / n1! n2! n3!...nm! ) a1n1 a1n1 ...... a1n1
Where n1 , n2 , n3 ,...nm are non-negative integers such that n1 + n2 + n3 + .... + nm = n
(a) The coefficients of a1n1 a2n2 ...... amn1 in the expansion of (a1 + a2 + a3 +....+am )n is (n! / n1! n2! n3!...nm! )
(b) The number of dissimilar terms in the expansion of (a1 + a2 + a3 +....+am )n = n+m-1Cm-1
7. Binomial theorem for negative or fractional index
When n is negative and/or fraction and |x|<1 then
(1+x)n = 1 + nx + ( (n(n-1) ) / 2! ) x2 + ( (n(n-1)(n-2)) / 3! ) x3 + .... + ( (n(n-1)(n-2).....(n-r+1)) / r! ) xr + ...........upto infinity
The general term is
Tr+1 = ( (n(n-1)(n-2).....(n-r+1)) / r! ) xr
For example, if |x|<1 then,
(a) (1+x)-1 = 1 - x + x2 - x3 + x4 - .......
(b) (1 - x)-1 = 1 + x + x2 + x3 + x4 - .......
(c) (1+x)-2 = 1 - 2x + 3x2 - 4x3 + 5x4 - .......
(d) (1 - x)-1 = 1 + 2x + 3x2 + 4x3 + 5x4 - .......
(e) (1 + x)(1/2) = 1 + (1/2)x + ( (1/2)(-1/2) / 2! )x2 + ( (1/2)(-1/2)(-3/2) / 3! )x3 +......
(f) (1 + x)(-1/2) = 1 - (1/2)x + ( (1/2)(3/2) / 2! )x2 + ( (1/2)(3/2)(-3/2) / 3! )x3 +......
8. Using Differentiation and Integration in Binomial Theorem
(a) Whenever the numerical occur as a product of binomial coefficients, differentiation is useful.
(b) Whenever the numerical occur as a fraction of binomial coefficients, integration is useful.
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