Study Notes for CAT 2021: Binomial Theorem

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial

(x + y) n

into a sum involving terms of the form

ax b y c

, where the exponents

b

and

c

are nonnegative integers with

b + c = n

, and the coefficient

a

of each term is a specific positive integer depending on

n

and

b

1. Binomial Theorem for positive integral index

If n is a positive integer and x, y is real or imaginary then (x+y)ⁿ= ⁿC₀x^n-0 y⁰ + ⁿC₁x^n-1 y¹ + ..... +ⁿC_rx^n-r y^r + ......ⁿC_nx⁰ y^r , i.e (x+y)ⁿ= ∑ ⁿC₀x^n-r y^r

General term is T_r+1 = ⁿC_rx^n-r y^rHere ⁿC_{0 ,} ⁿC_{1 ,}ⁿC_{r ,}ⁿC_n are called the binomial coefficients.

Replacing y by -y the general term of (x-y)ⁿ is obtained as

T_r+1 = (-1)^r ⁿC_rx^n-r y^r

Similarly, the general term of (1 + x)ⁿ and (1 - x)ⁿcan be obtained by replacing x by 1 and x by 1 and y by -x respectively.

ⁿC_r = ⁿC_(n-_r)

ⁿC_r + ⁿC_(n-_r) = ⁿ⁺¹C_r

ⁿC_r / ⁿC_r-1 = (n - r + 1) / r

r (ⁿC_r) = n ( ^n-1C_r-1 )

ⁿC_r / (r+1) = ( ⁿ⁺¹C_r+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 )^th^h 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 T_r and T_r₊₁ be the r^th and (r+1)^th terms in the expansion, (x+y)ⁿ then

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

Let T_r+1 be the numerically greatest term in the above expression

Then

T_r+1≥T_r

(( 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 ⁿC_{0 ,} ⁿC_{1 ,}ⁿC_{r ,}ⁿC_n respectively

Since (1+x)ⁿ = ⁿC₀ + ⁿC₁x + ⁿC₂x² + ..... +ⁿC_nxⁿ

Put x=1

ⁿC₀ + ⁿC₁+ ⁿC₂ + ..... +ⁿC_n=2ⁿ

Put x=-1

ⁿC₀ + ⁿC₂ + ⁿC₄+.....=ⁿC₁+ ⁿC₃+ ⁿC₅

Sum of odd terms coefficients=Sum of even terms coefficients=2^n-1

5. Series Summation

ⁿC₀ + ⁿC₁+ ⁿC₂ + ..... +ⁿC_n=2ⁿ

(b) ⁿC₀ - ⁿC₁ +ⁿC₂ - ⁿC₃ + ... + (-1)ⁿ ⁿC_n = 0

(c) ⁿC₀ + ⁿC₂ + ⁿC₄+.....=ⁿC₁+ ⁿC₃+ ⁿC₅= 2^n-1

(d) ⁿC²₀ + ⁿC²₁+ ⁿC²₂ + ..... +ⁿC²_n=(2n) ! /(n!)³

6. Multinomial Expansion

If n is a positive integer a₁ , a₂ ,... a_m are real or imaginary then

(a₁ + a₂ + a₃ +....+a_m )ⁿ = ∑(n! / n₁! n₂! n₃!...n_m! ) a₁^n₁a1ⁿ¹...... a₁^n₁

Where n₁ , n₂, n₃,...n_m are non-negative integers such that n₁+ n₂+ n₃+ .... + n_m = n

(a) The coefficients of a₁^n₁a₂^n₂...... a_m^n₁in the expansion of (a₁ + a₂ + a₃ +....+a_m )ⁿ is (n! / n₁! n₂! n₃!...n_m! )

(b) The number of dissimilar terms in the expansion of (a₁ + a₂ + a₃ +....+a_m )ⁿ = ^n+m-1C_m-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! ) x^r + ...........upto infinity

The general term is

T_r+1 = ( (n(n-1)(n-2).....(n-r+1)) / r! ) x^r

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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