The value of (nCo/n)+(nC1/n+1)+(nC2/n+2)+. . .+(nCn/2n) is equal to

Question

The value of (nCo/n)+(nC1/n+1)+(nC2/n+2)+. . .+(nCn/2n) is equal to (A) ∫ limits1oxn-1(1-x)n dx (B) ∫ limits21xn(x-1)n-1 dx  (C) ∫ limits21xn - 1(1-x)n dx  (D) ∫ limits 10(1-x)n xn-1 dx

Tardigrade Admin · Accepted Answer

S=(nC0/n)+(nC1/n+1)+(nC2/n+2)+...+(nCn/2n) =((1/x))r= mCr x2m-3r nC1 ∫ limits01 x n dx + ... + nCn ∫ limits01 x2n-1 dx =∫ limits01 [nC0x n-1+ nC1xn+...+ nCnx2n-1]dx =((1/x))r= mCrx2m-3r =∫ limits12 xn(x-1)n-1dx

Q. The value of $\frac{^{n} C _{o}}{n} + \frac{^{n} C _{1}}{n + 1} + \frac{^{n} C _{2}}{n + 2} + ... + \frac{^{n} C _{n}}{2 n}$ is equal to

$o \int 1 x^{n - 1} (1 - x)^{n} d x$

$1 \int 2 x^{n} (x - 1)^{n - 1} d x$

$1 \int 2 x^{n - 1} (1 - x)^{n} d x$

$0 \int 1 (1 - x)^{n} x^{n - 1} d x$

Q. The value of nnCo​​+n+1nC1​​+n+2nC2​​+...+2nnCn​​ is equal to

o∫1​xn−1(1−x)ndx

1∫2​xn(x−1)n−1dx

1∫2​xn−1(1−x)ndx

0∫1​(1−x)nxn−1dx

S=nnC0​​+n+1nC1​​+n+2nC2​​+...+2nnCn​​ =(x1​)r=mCr​x2m−3rnC1​0∫1​xndx+...+nCn​0∫1​x2n−1dx =0∫1​[nC0​xn−1+nC1​xn+...+nCn​x2n−1]dx =(x1​)r=mCr​x2m−3r =1∫2​xn(x−1)n−1dx

Q. The value of $\frac{^{n} C _{o}}{n} + \frac{^{n} C _{1}}{n + 1} + \frac{^{n} C _{2}}{n + 2} + ... + \frac{^{n} C _{n}}{2 n}$ is equal to

$o \int 1 x^{n - 1} (1 - x)^{n} d x$

$1 \int 2 x^{n} (x - 1)^{n - 1} d x$

$1 \int 2 x^{n - 1} (1 - x)^{n} d x$

$0 \int 1 (1 - x)^{n} x^{n - 1} d x$