Binomial Theorem Notes for IIT JEE, Download PDF!
JEE Main Short Notes
Binomial theorem is an important and basic formula in algebra. It has its applications in other topics as well. Thus, it is very important for a JEE Main aspirant to prepare this topic in a well-versed manner. These notes are also useful in your JEE Advanced and BITSAT preparation. You can also download Binomial Theorem Notes PDF for IIT JEE preparation at the end of the post.
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
Moreover, you can also check the syllabus for JEE Main exam through the links shared below:
All the best!
Download Goprep, the best IIT JEE Preparation App