If `C_r=(n !)/([r !(n-r)]),` the prove that `sqrt(C_1)+sqrt(C_2)+.......sqrt(C_n) lt sqrt(n(2^n-1)) ="">

If C_(r ) = (n!)/(r!(n - r)!) , then prove that sqrt(C_(1)) + sqrt(C_(2)) + …. + sqrt(C_(n)) sqrt(n(2^(n) - 1))

If (1+x)^(n)=sum_(r=0)^(n)C_(r)x^(r) then prove that C_(0)+(C_(1))/(2)+......+(C_(n))/(n+1)=2

If (1+x)^(n)=sum_(r=0)^(n)C_(r)x^(r) then prove that C_(1)+2C_(2)+3C_(3)+....+nC_(n)=n2^(n-1)

If (1+x)^(n)=sum_(r=0)^(n)C_(r)x^(r), then prove that C_(1)+2c_(2)+3C_(1)+...+nC_(n)=n2^(n-1)...

Prove that nC_(r)+n-1C_(r)+n-2C_(r)+.......+rC_(r)=n+1C_(1)

lim_ (n rarr oo) sum_ (n = 1) ^ (n) (sqrt (n)) / (sqrt (r) (3sqrt (r) + 4sqrt (n)) ^ (2))

The value of underset( n rarroo)(lim) underset(r=1)overset(r=4n)(sum)(sqrt(n))/(sqrt(r ) (3 sqrt(r)+4)sqrt(n)^(2)) is equal to