该数列的通项为(2k+1)√(2k-1)+(2k-1)√(2k+1).
对任意的k=1,2,...,25有
(2k+1)√(2k-1)+(2k-1)√(2k+1)
=((2k+1)√(2k-1)-(2k-1)√(2k+1))/(2(2k-1)(2k+1))
=(√(2k-1)/(2k-1)-√(2k+1)/(2k+1))/2
=(1/√(2k-1)-1/√(2k+1))/2
于是得
1/(3+√3)+1/(5√3+3√5)+1/(7√5+5√7)+...+1/(49√47+47√49)
=(1/√1-1/√3)/2+(1/√3-1/√5)/2+...+(1/√47-1/√49)/2
=(1/√1-1/√3+1/√3-1/√5+...+1/√47-1/√49)/2
=(1-1/√49)/2=3/7