Question

(1除以2根号1加根号2)加（1除以3根号2加2根号3）加----加1除以100根号99加99根号100

Answer 1

通式是 1/[(a+1)√a + a √(a+1)]

= [(a+1)√a - a √(a+1)] / { [(a+1)√a - a √(a+1)] [(a+1)√a+a √(a+1)] }

= [(a+1)√a - a √(a+1)]/ [(a+1)^2 a - a^2 (a+1)]

= √(a(a+1) ) [√(a+1)- √a]/ {a(a+1) [ a+1-a]}

= [√(a+1)- √a]/ √ [a(a+1)]

= 1/ √a - 1/ √(a+1)

所以得 1-1/√2+ 1/√2-1/√3 +1/√3-1/√4 +.- 1/√(100)

= 1- 1/√(100)

= 9/10