JEE Main Short Notes

By Neha BagriUpdated : Sep 10, 2020 , 7:39 IST

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

Then

Tr+1≥Tr

Or

(( n - r + 1 ) / r ) x

Greatest Coefficient:

(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)

## 4.Binomial Coefficients

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

Put x=1

nC0 + nC1 + nC2 + ..... +nCn =2n

Put x=-1

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:

JEE Main Syllabus with weightage

JEE Main Question Paper 2019 with Solutions

All the best!

Team Goprep

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