Solve this Interesting problem. don't give meaningless answers I will report.If you can't solve it don't answer it.
Answers
Step-by-step explanation:
Given :-
1/(1+√2) + 1/(√2+√3) +1/(√3+√4)+ ...
+ 1/(√8+√9)
To find :-
Find the value of the given numerical expression?
Solution :-
Given numerical expression is
1/(1+√2) + 1/(√2+√3) +1/(√3+√4)+ ...
+ 1/(√8+√9)
The denominator in 1/(1+√2) = 1+√2
The Rationalising factor of 1+√2 = 1-√2
On Rationalising the denominator of
1/(1+√2) then
=> [1/(1+√2)]×[(1-√2)/(1-√2)]
=> (1-√2)/[(1+√2)(1-√2)]
=> (1-√2)/[1²-(√2)²]
Since (a+b)(a-b) = a²-b²
Where , a = 1 and b = √2
=> (1-√2)/(1-2)
=> (1-√2)/(-1)
=> -(1-√2)
=> -1+√2
=> √2-1
1/(1+√2) = √2-1 --------------------(1)
The denominator in 1/(√2+√3) = √2+√3
The Rationalising factor of √2 +√3 is √2+√3
On Rationalising the denominator of
1/(√2+√3)
=> [1/(√2+√3)]×[(√2-√3)/(√2-√3)]
=> (√2-√3)/[(√2+√3)(√2-√3)]
=> (√2-√3)/[(√2)²-(√3)²]
Since (a+b)(a-b) = a²-b²
Where , a = √2 and b = √3
=> (√2-√3)/(2-3)
=> (√2-√3)/(-1)
=> -(√2-√3)
=> -√2+√3
=> √3-√2
1/(√2+√3) = √3-√2 --------------------(2)
The denominator in 1/(√3+√4) = √3+√4
The Rationalising factor of √3+√4 is √3-√4 then
On Rationalising the denominator of
1/(√3-√4)
=> [1/(√3+√4)]×[(√3-√4)/(√3-√4)]
=> (√3-√4)/[(√3+√4)(√3-√3)]
=> (√3-√4)/[(√3)²-(√4)²]
Since (a+b)(a-b) = a²-b²
Where , a = √3 and b = √4
=> (√3-√4)/(3-4)
=> (√3-√4)/(-1)
=> -(√3-√4)
=> -√3+√4
=> √4-√3
1/(√3+√4) = √4-√3 --------------------(3)
and the same way
The denominator in 1/(√8+√9) = √8+√9
The Rationalising factor of √8+√9 is √8-√9
On Rationalising the denominator of
1/(√8+√9)
=> [1/(√8+√9)]×[(√8-√9)/(√8-√9)]
=> (√8-√9)/[(√8+√9)(√8-√9)]
=> (√8-√9)/[(√8)²-(√9)²]
Since (a+b)(a-b) = a²-b²
Where , a = √8 and b = √9
=> (√8-√9)/(8-9)
=> (√8-√9)/(-1)
=> -(√8-√9)
=> -√8+√9
=> √9-√8
1/(√8+√9) = √9-√8 --------------------(4)
On adding all equations then we get
=>√2-1+√3-√2+√4-√3+√5-√4+√6-√5
+√7-√6+√8-√7+√9-√8
=>-1 + (√2-√2)+(√3-√3)+(√4-√4)+(√5-√5) +(√6-√6)+(√7-√7)+(√8-√8)+√9
=> -1+√9
=> -1+3
=> 3-1
=> 2
Answer:-
The value of the given expression for the given problem is 2
Used formulae:-
- The Rationalising factor of √a+b is √a-b
- The Rationalising factor of √a+√b is √a -√b
- (a+b)(a-b) = a²-b²