Binomial Theorem and Mathematical Induction Notes Class 11th Maths

Binomial Theorem for Positive Integer

If n is any positive integer, then

This is called binomial theorem.

Here, ⁿC₀, ⁿC₁, ⁿC₂, … , ⁿn_o are called binomial coefficients and

ⁿC_r = n! / r!(n – r)! for 0 ≤ r ≤ n.

Properties of Binomial Theorem for Positive Integer

(i) Total number of terms in the expansion of (x + a)ⁿ is (n + 1).

(ii) The sum of the indices of x and a in each term is n.

(iii) The above expansion is also true when x and a are complex numbers.

(iv) The coefficient of terms equidistant from the beginning and the end are equal. These coefficients are known as the binomial coefficients and

ⁿC_r = ⁿC_{n – r}, r = 0,1,2,…,n.

(v) General term in the expansion of (x + c)ⁿ is given by

T_{r + 1} = ⁿC_rx^{n – r} a^r.

(vi) The values of the binomial coefficients steadily increase to maximum and then steadily decrease .

(vii)

(viii)

(ix) The coefficient of x^r in the expansion of (1+ x)ⁿ is ⁿC_r.

(x)

(xi) (a)

(b)

(xii) (a) If n is odd, then (x + a)ⁿ + (x – a)ⁿ and (x + a)ⁿ – (x – a)ⁿ both have the same number of terms equal to (n +1 / 2).

(b) If n is even, then (x + a)ⁿ + (x – a)ⁿ has (n +1 / 2) terms. and (x + a)ⁿ – (x – a)ⁿ has (n / 2) terms.

(xiii) In the binomial expansion of (x + a)ⁿ, the r th term from the end is (n – r + 2)th term fromthe beginning.

(xiv) If n is a positive integer, then number of terms in (x + y + z)ⁿ is (n + l)(n + 2) / 2.

Middle term in the Expansion of (1 + x)ⁿ

(i) It n is even, then in the expansion of (x + a)ⁿ, the middle term is (n/2 + 1)^th terms.

(ii) If n is odd, then in the expansion of (x + a)ⁿ, the middle terms are (n + 1) / 2 th term and (n + 3) / 2 th term.

Greatest Coefficient

(i) If n is even, then in (x + a)ⁿ, the greatest coefficient is ⁿC_{n / 2}

(ii) Ifn is odd, then in (x + a)ⁿ, the greatest coefficient is ⁿC_{n – 1 / 2} or ⁿC_{n + 1 / 2} both being equal.

Greatest Term

In the expansion of (x + a)ⁿ

(i) If n + 1 / x/a + 1 is an integer = p (say), then greatest term is T_p == T_{p + 1}.

(ii) If n + 1 / x/a + 1 is not an integer with m as integral part of n + 1 / x/a + 1, then T_{m + 1}. is the greatest term.

Important Results on Binomial Coefficients

Divisibility Problems

From the expansion, (1+ x)ⁿ = 1+ ⁿC₁x + ⁿC₁x²+ … +ⁿC_nxⁿ

We can conclude that,

(i) (1+ x)ⁿ – 1 = ⁿC₁x + ⁿC₁x²+ … +ⁿC_nxⁿ is divisible by x i.e., it is multiple of x.

(1+ x)ⁿ – 1 = M(x)

(ii)

(iii)

Multinomial theorem

For any n ∈ N,

(i)

(ii)

(iii) The general term in the above expansion is

(iv)The greatest coefficient in the expansion of (x₁ + x₂ + … + x_m)ⁿ iswhere q and r are the quotient and remainder respectively, when n is divided by m.

(v) Number of non-negative integral solutions of x₁ + x₂ + … + x_n = n is ^{n + r – 1}C_{r – 1}

R-f Factor Relations

Here, we are going to discuss problem involving (√A + B)sup>n = I + f, Where I and n are positive integers.

0 le; f le; 1, |A – B²| = k and |√A – B| < 1

Binomial Theorem for any Index

If n is any rational number, then

(i) If in the above expansion, n is any positive integer, then the series in RHS is finite otherwise infinite.

(ii) General term in the expansion of (1 + x)ⁿ is T_{r + 1 = n(n – 1)(n – 2)… [n – (r – 1)] / r! * x^r}

(iii) Expansion of (x + a)ⁿ for any rational index

(vii) (1 + x)^{– 1} = 1 – x + x² – x³ + …∞

(viii) (1 – x)^{– 1} = 1 + x + x² + x³ + …∞

(ix) (1 + x)^{– 2} = 1 – 2x + 3x² – 4x³ + …∞

(x) (1 – x)^{– 2} = 1 + 2x + 3x² – 4x³ + …∞

(xi) (1 + x)^{– 3} = 1 – 3x + 6x² – …∞

(xii) (1 – x)^{– 3} = 1 + 3x + 6x² – …∞

(xiii) (1 + x)ⁿ = 1 + nx, if x², x³,… are all very small as compared to x.

Important Results

(i) Coefficient of x^m in the expansion of (ax^p + b / x^q)ⁿ is the coefficient of T_{r + l} where r = np – m / p + q

(ii) The term independent of x in the expansion of ax^p + b / x^q)ⁿ is the coefficient of T_{r + l}where r = np / p + q

(iii) If the coefficient of rth, (r + l)th and (r + 2)th term of (1 + x)ⁿ are in AP, then n² – (4r+1) n + 4r² = 2

(iv) In the expansion of (x + a)ⁿ

T_{r + 1} / T_r = n – r + 1 / r * a / x

(v) (a) The coefficient of x^{n – 1} in the expansion of

(x – l)(x – 2) ….(x – n) = – n (n + l) / 2

(b) The coefficient of x^{n – 1} in the expansion of

(x + l)(x + 2) ….(x + n) = n (n + l) / 2

(vi) If the coefficient of pth and qth terms in the expansion of (1 + x)ⁿ are equal, then p + q = n + 2

(vii) If the coefficients of x^r and x^{r + 1} in the expansion of a + x / b)ⁿ are equal, then

n = (r + 1)(ab + 1) – 1

(viii) The number of term in the expansion of (x₁ + x₂ + … + x_r)_{n is n + r – 1C r – 1.}

(ix) If n is a positive integer and a₁, a₂, … , a_m ∈ C, then the coefficient of x^r in the expansion of (a₁ + a₂x + a₃x² +… + a_mx^{m – 1})ⁿ is

(x) For |x| < 1,

(a) 1 + x + x² + x³+ … + ∞ = 1 / 1 – x

(b) 1 + 2x + 3x² + … + ∞ = 1 / (1 – x)²

(xi) Total number of terms in the expansion of (a + b + c + d)ⁿ is (n + l)(n + 2)(n + 3) / 6.

Important Points to be Remembered

(i) If n is a positive integer, then (1 + x)ⁿ contains (n +1) terms i.e., a finite number of terms. When n is general exponent, then the expansion of (1 + x)ⁿ contains infinitely many terms.

(ii) When n is a positive integer, the expansion of (l + x)ⁿ is valid for all values of x. If n is general exponent, the expansion of (i + x)ⁿ is valid for the values of x satisfying the condition |x| < 1.