Binomial theorm for a +ve integral index
If n is a +ve integer and x, y are two complex numbers then ,
(x + y)n = nCoXn + nC1 Xn-1 y + nC2 Xn-2 y2 +.....+nCr xn-r yr + .....
.......+ n Cn-1 x yn-1 + nCn yn .
The coefficients n Co, nC1, .................nCn 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 nCr = nCn-r .
(iv) The term nCr xn-r yr is the (r+1)th term from the beginning of the expansion. it is usually denoted by Tr +1 and is called the general term of the expansion.
Middle term
if n is even , then the expansion (x + y)n has just one middle term i.e.
(n/2 +1)th term. it is given by nCn/2 xn/2 yn/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)n is nCn / 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....
| tr-1 | < | tr | and | tr +1 | < | tr | to obtain some desired results.
Some other useful expansions.
1. (x - y)n = nCo Xn - nC1 Xn-1 y + nC2 Xn-2 y2 +.....+ (-1) n n Cn yn
2. (x + y)n + (x - y)n = 2( nCoXn + nC2 Xn-2 y2 + nC4 Xn-4 +......)
3. (x + y)n - (x - y)n = 2( nCoXn + nC3 Xn-3 y3 + nC5 Xn-5 +......)
4. (1 + x)n + (1 - x)n = 2( nCo + nC2 X2 + nC4 Xn-4 +......)
5. (1 + x)n - (1 - x)n = 2( nCo + nC1 X + nC3 X3 +......)
6. nC1 + 2nC2 X + 3 nC3 X2 +......+ n Cn xn-1 = n(1+x)n-1
Properties of the binomial coefficients
1.nC0 + nC1 +nC2 + nC3 +......+ n Cn = 2n
2.nC0 + nC2 +nC4 + nC6 +......+ n Cn = 2n-1
3. nC0 - nC1 +nC2 - nC3 +......+(-1)n-1 n Cn = 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 nCr-1 , nCr , nCr+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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