Binomial Theorem for Positive Integer

If n is any positive integer, then

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

This is called binomial theorem.

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

nCr = 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

nCr = nCn – r, r = 0,1,2,…,n.

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

Tr + 1 = nCrxn – r ar.

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

(vii) CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(viii) CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(ix) The coefficient of xr in the expansion of (1+ x)n is nCr.

(x)

(xi) (a) CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(b) CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(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 from CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Inductionthe 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 nCn / 2

(ii) Ifn is odd, then in (x + a)n, the greatest coefficient is nCn – 1 / 2 or nCn + 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 Tp == Tp + 1.

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

Important Results on Binomial Coefficients

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction
CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

Divisibility Problems

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

We can conclude that,

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

(1+ x)n – 1 = M(x)

(ii) CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(iii) CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

Multinomial theorem

For any n ∈ N,

(i)CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(ii) CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(iii) The general term in the above expansion is

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(iv)The greatest coefficient in the expansion of (x1 + x2 + … + xm)n is CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction where 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 + … + xn = n is n + r – 1Cr – 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

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(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 Tr + 1 = n(n – 1)(n – 2)… [n – (r – 1)] / r! * xr

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

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(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 xm in the expansion of (axp + b / xq)n is the coefficient of Tr + l where r = np – m / p + q

(ii) The term independent of x in the expansion of axp + b / xq)n is the coefficient of Tr + 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

Tr + 1 / Tr = n – r + 1 / r * a / x

(v) (a) The coefficient of xn – 1 in the expansion of

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

(b) The coefficient of xn – 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 xr and xr + 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 + … + xr)n is n + r – 1C r – 1.

(ix) If n is a positive integer and a1, a2, … , am ∈ C, then the coefficient of xr in the expansion of (a1 + a2x + a3x2 +… + amxm – 1)n is

CBSE Class 11 Maths Notes Binomial Theorem and Mathematical Induction

(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.

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