Prove that - 1-2-1 - 1-3-2- 1-4-3-1-5-4-1-6-5-1-7-6-1-8-7 -1-9-8 -2

LHS =1/√(2)+1+ 1/√(2)+√(2)+1/√(4)+√(3)+.....+ 1/√(8)+√(7)+ 1/√(9)+√(8) Rationalizing each term: ⇒√(2) 1/1+√(3) √(2)/1+√(4) √(3)/1+...+√(8) √(7)+√(9) √(8 ...

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Standard X

Mathematics

Number Theory: Interesting Results

Prove that : ...

Question

Prove that : $\frac{1}{\sqrt{2} + 1} + \frac{1}{\sqrt{3} + \sqrt{2}} + \frac{1}{\sqrt{4} + \sqrt{3}} + \frac{1}{\sqrt{5} + \sqrt{4}} + \frac{1}{\sqrt{6} + \sqrt{5}} + \frac{1}{\sqrt{7} + \sqrt{6}} + \frac{1}{\sqrt{8} + \sqrt{7}}$
$+ \frac{1}{\sqrt{9} + \sqrt{8}} = 2$

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Solution

LHS $= \frac{1}{\sqrt{2} + 1} + \frac{1}{\sqrt{2} + \sqrt{2}} + \frac{1}{\sqrt{4} + \sqrt{3}} + . . . . . + \frac{1}{\sqrt{8} + \sqrt{7}} + \frac{1}{\sqrt{9} + \sqrt{8}}$

Rationalizing each term:

$\Rightarrow \frac{\sqrt{2} - 1}{1} + \frac{\sqrt{3} - \sqrt{2}}{1} + \frac{\sqrt{4} - \sqrt{3}}{1} + . . . + \sqrt{8} - \sqrt{7} + \sqrt{9} - \sqrt{8}$

$\Rightarrow 3 - 1 = 2$ Hence Proved

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Similar questions

Q. Find :

\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{4} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{8}} + \frac{1}{\sqrt{8} + \sqrt{9}}

$\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{4} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{8}} + \frac{1}{\sqrt{8} + \sqrt{9}} =$ __________.

Q. The value of

\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{4} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{8}} + \frac{1}{\sqrt{8} + \sqrt{9}}

is.

Q. The value of :

\frac{1}{\sqrt{9} + \sqrt{8}} + \frac{1}{\sqrt{8} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{4}} + \frac{1}{\sqrt{4} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{1}}

is :

Q. Study the following number line and answer the questions that follow.
2 5 9 1 7 2 5 1 9 7 3 9 5 2 4 6 8 1 9 7 5 2 7 3 1 9 7 8 5 2 6 5 4 1 9 7 8 2 0 1 0 9 7 8 5 1 9 7 3 5 1 6 9 7 2 1 7 3 7 9 5 1

How many times is number 5 followed by 1 or 2, but not preceded by 8 ?

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Prove that : 1√2+1+1√3+√2+1√4+√3+1√5+√4+1√6+√5+1√7+√6+1√8+√7+1√9+√8=2

LHS=1√2+1+1√2+√2+1√4+√3+.....+1√8+√7+1√9+√8Rationalizing each term:⇒√2−11+√3−√21+√4−√31+...+√8−√7+√9−√8⇒3−1=2 Hence Proved

Prove that : $\frac{1}{\sqrt{2} + 1} + \frac{1}{\sqrt{3} + \sqrt{2}} + \frac{1}{\sqrt{4} + \sqrt{3}} + \frac{1}{\sqrt{5} + \sqrt{4}} + \frac{1}{\sqrt{6} + \sqrt{5}} + \frac{1}{\sqrt{7} + \sqrt{6}} + \frac{1}{\sqrt{8} + \sqrt{7}}$
$+ \frac{1}{\sqrt{9} + \sqrt{8}} = 2$