Question

plz answer fast
I wI'll mark the correct answer as brainliest​

Answer 1

1/(root4 + root5) + 1/(root5 + root6) + 1/(root6 + root7) + 1/(root7 + root8) + 1/(root8 + root9) = 1

Step-by-step explanation:

In all, numbers are like: $\frac{1}{\sqrt{n} +\sqrt{n+1}}$

                  = $\frac{1}{\sqrt{n}+\sqrt{n+1}}$

                  = $\small{\frac{1}{\sqrt{n}+\sqrt{n+1}}\times\frac{\sqrt{n}-\sqrt{n+1}}{\sqrt{n}-\sqrt{n+1}} }$

                  = $\frac{\sqrt{n}-\sqrt{n+1}}{(\sqrt{n})^2-(\sqrt{n+1})^2}$

                  = $\frac{\sqrt{n}-\sqrt{n+1}}{n-(n+1)}$

                  = $\frac{\sqrt{n}-\sqrt{n+1}}{n-n-1}$

                  = $\frac{\sqrt{n}-\sqrt{n+1}}{-1}$

                  = $\small{-\sqrt{n}+\sqrt{n+1}}$

Therefore,

$\frac{1}{\sqrt{4}+\sqrt{5}}=-\sqrt4 +\sqrt5$

$\frac{1}{\sqrt{5}+\sqrt{6}}=-\sqrt5 +\sqrt6$

 Similarly with all, now it is:

⇒ (-√4 + √5) + (-√5 + √6) + (-√6 + √7) + (-√7 + √8) + (-√8 + √9)

⇒ -√4 + √5 - √5 + √6 - √6 + √7 - √7 + √8 - √8 + √9

⇒ - √4 + √9

⇒ - 2 + 3

⇒ 3 - 2

⇒ 1  

   Proved

Answer 2

$\frac{ 1}{√4+√5 } + \frac{ 1}{√5+√6 } + \frac{ 1}{√6+√7 } + \frac{ 1}{√7+√8 } + \frac{ 1}{√ 8+√9 } + = 1$

$\frac{ 1}{√4+√5 } \times \frac{√4 - √5 }{√4 - √5 } = \frac{√4 + √5}{{(√4 - √5) }^{2} }}$

$\frac{ 1}{√4+√5 } \times \frac{√4 - √5 }{√4 - √5 } = \frac{√4 - √5}{ - 1} = - √4 + √5$

$\frac{ 1}{√5+√6 } \times \frac{√5 - √6 }{√5 - √6 } = \frac{√5 + √6}{{(√5 - √6) }^{2}}$

$\frac{ 1}{√5+√6 } \times \frac{√5 - √6 }{√5 - √6 } = \frac{√5 - √6}{ - 1} = - √5 + √6$

$\frac{ 1}{√6+√7 } \times \frac{√6 - √7 }{√6 - √7 } = \frac{√6+ √7}{{(√6 - √7) }^{2} }$

$\frac{ 1}{√6+√7 } \times \frac{√6 - √7 }{√6 - √7 } = \frac{√6 - √7}{ - 1} = - √6 + √7$

$\frac{ 1}{√7+√8 } \times \frac{√7- √8 }{√7 - √8 } = \frac{√7+ √8}{{(√7 - √8) }^{2} }$

$\frac{ 1}{√7+√8 } \times \frac{√7 - √8 }{√7 - √8 } = \frac{√7 - √8}{ - 1} = - √7 + √8$

$\frac{ 1}{√8+√9 } \times \frac{√8 - √9 }{√8 - √9 } = \frac{√8 + √9}{{(√8 - √9) }^{2} }$

$\frac{ 1}{√8+√9 } \times \frac{√8 - √9 }{√8 - √9 } = \frac{√8 - √9}{ - 1} = - √8 + √9$

$- √4+√5 - √5+√6 - √6+√7 - √7+√8 - √8+√9 = 1$

$- √4+√9 = 1$

Putting value of √4 and √9:-

$- 2 + 3 = 1$

LHS = RHS

Hence proved.