CBSE Class 11 Maths Binomial Theorem and Mathematical Induction – Get here the Notes for Class 11 Binomial Theorem and Mathematical Induction. Candidates who are ambitious to qualify the Class 11 with good score can check this article for Notes. This is possible only when you have the best CBSE Class 11 Maths study material and a smart preparation plan. To assist you with that, we are here with notes. Hope these notes will helps you understand the important topics and remember the key points for exam point of view. Below we provided the Notes of CBSE Class 11 Maths for topic Binomial Theorem and Mathematical Induction.

**Class:**11th**Subject:**Maths**Topic:**Binomial Theorem and Mathematical Induction**Resource:**Notes

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

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**Binomial Theorem for Positive Integer**

If n is any positive integer, then

This is called binomial theorem.

Here, ^{n}C_{0}, ^{n}C_{1}, ^{n}C_{2}, … , ^{n}n_{o} are called binomial coefficients and

^{n}C_{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

^{n}C_{r} = ^{n}C_{n – r}, r = 0,1,2,…,n.

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

T_{r + 1} = ^{n}C_{r}x^{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 ^{n}C_{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 from the 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 ^{n}C_{n / 2}

(ii) Ifn is odd, then in (x + a)^{n}, the greatest coefficient is ^{n}C_{n – 1 / 2} or ^{n}C_{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+ ^{n}C_{1}x + ^{n}C_{1}x^{2}+ … +^{n}C_{n}x^{n}

We can conclude that,

(i) (1+ x)^{n} – 1 = ^{n}C_{1}x + ^{n}C_{1}x^{2}+ … +^{n}C_{n}x^{n} 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 (x_{1} + x_{2} + … + x_{m})^{n} is where q and r are the quotient and remainder respectively, when n is divided by m.

(v) Number of non-negative integral solutions of x_{1} + x_{2} + … + 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 – B^{2}| = 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! * xr}

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

(vii) (1 + x)^{– 1} = 1 – x + x^{2} – x^{3} + …∞

(viii) (1 – x)^{– 1} = 1 + x + x^{2} + x^{3} + …∞

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

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

(xi) (1 + x)^{– 3} = 1 – 3x + 6x^{2} – …∞

(xii) (1 – x)^{– 3} = 1 + 3x + 6x^{2} – …∞

(xiii) (1 + x)^{n} = 1 + nx, if x^{2}, x^{3},… 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 n^{2} – (4r+1) n + 4r^{2} = 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 (x_{1} + x_{2} + … + x_{r})_{n is n + r – 1C r – 1.}

(ix) If n is a positive integer and a_{1}, a_{2}, … , a_{m} ∈ C, then the coefficient of x^{r} in the expansion of (a_{1} + a_{2}x + a_{3}x^{2} +… + a_{m}x^{m – 1})^{n} is

(x) For |x| < 1,

(a) 1 + x + x^{2} + x^{3}+ … + ∞ = 1 / 1 – x

(b) 1 + 2x + 3x^{2} + … + ∞ = 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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