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sneha kulkarni

Binomial theorm for a +ve integral index

If n is a +ve integer and x_, y are two complex numbers then ,
(x + y)ⁿ= ⁿC_oXⁿ + ⁿC₁ X^n-1 y + ⁿC₂ X^n-2 y² +.....+ⁿC_r x^n-r y^r+ .....

.......+ ⁿC_n-1x y^n-1+ ⁿC_n y^{n .}

The coefficients ⁿ C_o, ⁿC₁, .................ⁿC_n are called Binomial coefficients.

Properties of the binomial expansion

(i) There are (n + 1) terms in the expression .

(ii) in any term of the expansion (x + y)ⁿ, the sum of exponents of x and y is alwayz n .

(iii) The binomial coefficent of the terms equidistant from beginning and the end are equal since ⁿC_r = ⁿC_{n-r .}
(iv) The term ⁿC_r x^n-r y^ris the (r+1)th term from the beginning of the expansion. it is usually denoted by T_{r +1}and is called the general term of the expansion.

Middle term

if n is even , then the expansion (x + y)ⁿhas just one middle term i.e.

(n/2 +1)th term. it is given by ⁿC_n/2 x^n/2 y^n/2.

if n is odd , the the expansion has two middle terms i.e. (n+1)/2 th term and

{ (n+1)/2 +1}th term. these are given by ⁿC_{(n-1) / 2} x^{(n-1) / 2} y^{(n+1) / 2}

and ⁿC_{(n+1) / 2} x^{(n+1) / 2} y^{(n-1) / 2}

The greatest coefficient

if n is even , the coefficient in the expansion of (x + y)ⁿis ⁿC_{n / 2 .}

if n is odd , there are two greatest coefficients in the expansion of (x + y)^n . these are ⁿC_{(n+1) / 2}and ⁿC_{(n-1) / 2.}

TIP
if we are gievn tat tr is a numerically the greatest term , we use....

| t_r-1| < | t_r| and | t_{r +1}| < | t_r| to obtain some desired results.

Some other useful expansions.

1. (x - y)ⁿ = ⁿCo Xⁿ - ⁿC₁ X^n-1 y + ⁿC₂ X^n-2 y² +.....+ (-1) ⁿ ⁿC_n yⁿ

2. (x + y)ⁿ+ (x - y)ⁿ= 2( ⁿC_oXⁿ + ⁿC₂ X^n-2 y2 + ⁿC₄ X^n-4 +......)

3. (x + y)^n - (x - y)ⁿ= 2( ⁿC_oXⁿ + ⁿC₃ X^n-3 y3 + ⁿC₅ X^n-5 +......)
4. (1 + x)ⁿ + (1 - x)ⁿ= 2( ⁿC_o + ⁿC₂ X² + ⁿC₄ X^n-4 +......)

5. (1 + x)ⁿ - (1 - x)ⁿ= 2( ⁿC_o + ⁿC₁ X + ⁿC₃ X³ +......)
6. ⁿC₁ + 2ⁿC₂ X + 3 ⁿC₃ X² +......+ ⁿC_n x^n-1= n(1+x)^n-1

Properties of the binomial coefficients

1.ⁿC₀ + ⁿC₁ +ⁿC₂ + ⁿC₃ +......+ ⁿC_n = 2ⁿ

2.ⁿC₀ + ⁿC₂ +ⁿC₄ + ⁿC₆ +......+ ⁿC_n = 2^n-1

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

Some useful tips and tricks

1. to find remainder when xⁿ is divided by y, try to express x or some of its powers as ky(+-) 1. for instance, to find remainder wen 7²⁰⁰ is divided by 50, we write 7²⁰⁰ = 50-1 and use binomial theorem.

2. if a,b,r belong to Q and root r is an irrational number, then smetimes it is useful to switch from a+ b root r to a-b root r.

3. if p,q belong to Q and root p and root q are irrational then often it is preferable to simply (root p + root q)²ⁿ by first using (root p + root q)²= p+q+ 2 root(pq).

4. if three consecutive binomail coefficients ⁿC_r-1 , ⁿC_{r ,}ⁿC_r+1 are in AP., then r = 1/2 {n(+-) root(n+2)}

5. four consecutive binomail coefficients can never be in AP.

6. three consecutive binomail coeffcients can never be in GP or HP.

not copied and pasted...but typed wit hand...comment if useful...correct if wrong....

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