" If "u=v^(2)=w^(3)=z^(4)," prove that "log_(u)(uvwz)=1+(1)/(2)+(1)/(3)+(1)/(4)

If u=v^(2)=w^(3)=z^(4) ,then log_(u)(uvwz) is equal to

if a+ib=((u+i)^(2))/(2u-i) then prove that a^(2)+b^(2)=((u^(2)+1)^(2))/(4u^(2)+1)

If u = log ((x ^ (2) + y ^ (2)) / (x + y)), prove that x (u) / (x) + y (u) / (y) = 1

Let A={u,v,w,x} .A bijective function f:A rarr A is chosen at random, then the probability that f(u)!=u,f(v)!=v,f(w)!=w and f(x)!=x is 1) (3)/(8) 2) (11)/(24) 3) (1)/(64) 4) (7)/(24)

If z_(1) and z-2 are complex numbers and u=sqrt(z_(1)z_(2)), then prove that |z_(1)|+|z_(2)|=|(z_(1)+z_(2))/(2)+u|+|(z_(1)+z_(2))/(2)-u|

The function u_(n) takes on the following values : u_(1) = ( 1)/( 4) , u_(2) = ( 1)/( 4) + ( 1)/( 10)"…....." u_(n) =(1)/( 3+1) + ( 1)/( 3^(2) + 1)+"…........." (1)/( 3^(n) + 1)"…........." Prove that underset( n rarr oo) ( "lim") u_(a) lt (1)/(2)