Simplify the expression?: #1/(sqrt(144)+sqrt(145))+1/(sqrt(145)+sqrt(146))+...+1/(sqrt(168)+sqrt(169))#

Simplify the expression:

#1/(sqrt(144)+sqrt(145))+1/(sqrt(145)+sqrt(146))+...+1/(sqrt(168)+sqrt(169))#

1 Answer
George C.
Mar 23, 2017

#1#

Explanation:

First note that:

#1/(sqrt(n+1)+sqrt(n)) = (sqrt(n+1)-sqrt(n))/((sqrt(n+1)+sqrt(n))(sqrt(n+1)-sqrt(n))#

#color(white)(1/(sqrt(n+1)+sqrt(n))) = (sqrt(n+1)-sqrt(n))/((n+1)-n)#

#color(white)(1/(sqrt(n+1)+sqrt(n))) = sqrt(n+1)-sqrt(n)#

So:

#1/(sqrt(144)+sqrt(145))+1/(sqrt(145)+sqrt(146))+...+1/(sqrt(168)+sqrt(169))#

#=(sqrt(145)-sqrt(144))+(sqrt(146)-sqrt(145))+...+(sqrt(169)-sqrt(168))#

#=sqrt(169)-sqrt(144)#

#=13-12#

#=1#

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