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The value of (1)/(1+sqrt(2))+(1)/(sqrt(2...

The value of `(1)/(1+sqrt(2))+(1)/(sqrt(2)+sqrt(3))+(1)/(sqrt(3)+sqrt(4))+(1)/(sqrt(4)+sqrt(5))+(1)/(sqrt(5)+sqrt(6))+(1)/(sqrt(6)+sqrt(7))+(1)/(sqrt(7)+sqrt(8))+(1)/(sqrt(8)+sqrt(9))` is

A

0

B

1

C

2

D

4

Text Solution

Verified by Experts

The correct Answer is:
C
We have
`1/(1+sqrt(2))+1/(sqrt(2)+sqrt(3))+1/(sqrt(3)+sqrt(4))+1/(sqrt(4)+sqrt(5))`
`+1/(sqrt(5)+sqrt(6))+1/(sqrt(6)+sqrt(7))+1/(sqrt(7)+sqrt(8))+1/(sqrt(8)+sqrt(9))`
On rationalising each of the above number separately we get
`(1-sqrt(2))/(1-2)+(sqrt(2)-sqrt(3))/(2-3)+(sqrt(3)-sqrt(4))/(3-4)+(sqrt(4)-sqrt(5))/(4-5)`
`+(sqrt(5)-sqrt(6))/(5-6)+(sqrt(6)-sqrt(7))/(6-7)+(sqrt(7)-sqrt(8))/(7-8)+(sqrt(8)-sqrt(9))/(8-9)`
`=-(1-sqrt(2))-(sqrt(2)-sqrt(3))-(sqrt(3)-sqrt(4))-(sqrt(4)-sqrt(5))-(sqrt(5)-sqrt(6))`
`-(sqrt(6)-sqrt(7))-(sqrt(7)-sqrt(8))-(sqrt(8)-sqrt(9))`
`=-1+sqrt(9)=-1+3=2`
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