bino Flashcards
(1+x)^n= ?
nC0 + nC1x+nC2 x^2 ………. nCn x^n
what is the other form of nCr
nCn-r
(1-x)^n= ?
nC0 - nC1x+nC2 x^2 ……….(-1)^n nCn x^n
The general term or the (r+1)th term in the expansion of (x + y)n is given by?
Tr+1 =nCr x^n-r y^r
if n is even, there is only one middle term which is given by ?
T(n+2)/2= nCn/2. x ^n/2. y ^n/2
if n is odd, there are two middle terms which are?
T(n+1)/2 and T{(n+1)/2}+1
nCr will be maximum when ?
when r= n/2 if n is even
when r= n-1/2 or n+1/2 if n is odd
the sum of all binomial coefficients is
2^n
the sum of the binomial coefficients at odd position is equal to the sum of the binomial coefficients at ____ position and each is equal to _____
even, 2^n-1
C0 + C2 + C4 +………. = ?
C1 + C3 + C5 +……. = 2^ n-1
Write the steps in order to find the integral and fractional part of irrational numbers of the form (a + b root c)^ n
Step 1: Write the given expression equal to [N] + F, where [N] is its integral part and F is the fractional part.
Step 2: Define f by Replacing + sign in the given expression by - . So that if f always lies between 0 and 1.
Step 3 : Either add f to the expression in step 1 or subtract f from the expression in step 1 so that R.H.S is an integer.
Step 4: If f is added to the expression in step 1, then f + F will always come to out to be equal to 1 i.e., f = 1 - F
If f is subtracted from the expression in step 1, then f will always come to out to be equal to F.
Step 5: Obtain the value of the desired expression after getting F in the terms of f.
(1+ x) ^ -1 = ?
1- x+ x^2 - x^3+ x^4 - ………. till inf
(1- x) ^ -1 = ?
1+ x+ x^2 + x^3+ x^4 + ………. till inf
(1+ x) ^ -2 = ?
1- 2x+ 3x^2 - 4x^3 + ………. till inf
(1 + x)^n = ?
1 + nx + n(n-1) / 1.2 * x^2 + n(n-1) (n-2)/1.2.3 * x^3………
