If `n=12 m(m in N),` prove that `^n C_0-(^n C_2)/((2+sqrt(3))^2)+(^n C_4)/((2+sqrt(3))^4)-(^n C_6)/((2+sqrt(3))^6)+ddot=((2sqrt(2))/(1+sqrt(3)))^ndot`

For all n in N,1+(1)/(sqrt(2))+(1)/(sqrt(3))+(1)/(sqrt(4))++(1)/(sqrt(n))

lim_(n rarr oo)((sqrt(n+3)-sqrt(n+2))/(sqrt(n+2)-sqrt(n+1)))

lim_(n to oo)[(sqrt(n+1)+sqrt(n+2)+....+sqrt(2n))/(n sqrt((n)))]

lim_(n rarr4)(sqrt(2n+1)-3)/(sqrt(n-1)-sqrt(2))

Prove that (^(2n)C_0)^2+(^(2n)C_1)^2+(^(2n)C_2)^2-+(^(2n)C_(2n))^2-(-1)^n^(2n)C_ndot

the value of ((-1+sqrt(3)i)/(2))^(3n)+((-1-sqrt(3)i)/(2))^(3n)=

Prove that ((2n)!)/(2^(2n)(n!)^(2))<=(1)/(sqrt(3n+1)) for all n in N

Prove that (.^(n)C_(1))/(2) + (.^(n)C_(3))/(4) + (.^(n)C_(5))/(6) + "…." = (2^(n) - 1)/(n+1) .

If a_(1),a_(2),a_(3)... are in A.P then prove that (1)/(sqrt(a)_(1)+sqrt(a)_(2))(+)/(sqrt(a)_(2)+sqrt(a)_(3))+...+(1)/(sqrt(a)_(n-1)+sqrt(a)_(n))=(n-1)/(sqrt(a)_(n)+sqrt(a)_(1))