Question

prove that 1/√4+√5 + 1/√5+√6 +1/√6+√7 1/√7+√8 1/√8+√9=1

Answer 1

1 / (√4 + √5)
= [1 / (√4 + √5)] x [(√4 - √5)/(√4 - √5)]
= (√4 - √5) / (√4)2 - (√5)2
= (√4 - √5) / 4 - 5
= (√4 - √5) / (-1)
= (√5 - √4)

(1 / √5 + √6)

= [1 / (√5 + √6)] x [(√5 - √6)/(√5 - √6)]
= (√5 - √6) / (√5)2 - (√6)2
= (√5 - √6) / 5 - 6
= (√5 - √6) / (-1)
= (√6 - √5)

Similarly,
(1 / √6 + √7) = (√7 - √6)
(1 / √7 + √8) = (√8 - √7)
(1 / √8 + √9) = (√9 - √8)

(1 / √4 + √5) + (1 / √5 + √6) + (1 / √6 + √7) + (1 / √7 + √8) + (1 / √8 + √9)
= (√5 - √4) + (√6 - √5) + (√7 - √6) + (√8 - √7) + (√9 - √8)
= 3 - 2
= 1