Evaluate- 1-2-1-1-3-2-1-4-3-1-100-99

The correct option is B 9 1√(2)+1=√(2) 1(√(2)+1)(√(2) 1)=√(2) 12 1=√(2) 1 Similarly, 1√(3)+√(2)=√(3) √(2), 1√(4)+√(3)=√(4) √(3), 1√(99 ...

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Byju's Answer

Standard XII

Mathematics

Arithmetic Progression

Evaluate: 1...

Question

Evaluate: $\frac{1}{\sqrt{2} + 1} + \frac{1}{\sqrt{3} + \sqrt{2}} + \frac{1}{\sqrt{4} + \sqrt{3}} + . . . \frac{1}{\sqrt{100} + \sqrt{99}}$

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Solution

The correct option is B $9$
$\frac{1}{\sqrt{2} + 1} = \frac{\sqrt{2} - 1}{(\sqrt{2} + 1) (\sqrt{2} - 1)} = \frac{\sqrt{2} - 1}{2 - 1} = \sqrt{2} - 1$
Similarly, $\frac{1}{\sqrt{3} + \sqrt{2}} = \sqrt{3} - \sqrt{2},$
$\frac{1}{\sqrt{4} + \sqrt{3}} = \sqrt{4} - \sqrt{3},$
$\frac{1}{\sqrt{99} + \sqrt{98}} = \sqrt{99} - \sqrt{98},$
$\frac{1}{\sqrt{100} + \sqrt{99}} = \sqrt{100} - \sqrt{99}$

$∴$ Given exp. $= \sqrt{2} - 1 + \sqrt{3} - \sqrt{2} + \sqrt{4} - \sqrt{3} + . . . .$ $+ \sqrt{99} - \sqrt{98} + \sqrt{100} - \sqrt{99}$
$= \sqrt{100} - 1 = 10 - 1 = 9.$

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Similar questions

Evaluate : $(i) \frac{5}{9} + \frac{- 7}{6} (i i) 4 + \frac{3}{- 5} (i i i) \frac{1}{- 15} + \frac{5}{- 12} (i v) \frac{5}{9} + \frac{3}{- 4} (v) \frac{- 8}{9} + \frac{- 5}{12} (v i) 0 + \frac{- 2}{7} (v i i) \frac{5}{- 11} + 0 (v i i i) 2 + \frac{- 3}{5} (i x) \frac{4}{- 9} + 1$

Q. If

\frac{\frac{1}{\sqrt{9}} - \frac{1}{\sqrt{11}}}{\frac{1}{\sqrt{9}} + \frac{1}{\sqrt{11}}} \times \frac{10 + \sqrt{99}}{x} = \frac{1}{2}

then x equals

Q. The value of

\frac{1}{1 \times 2} + \frac{1}{2 \times 3} + \frac{1}{3 \times 4} + . . . . + \frac{1}{99 \times 100}

is ____.

Q. Find 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{99} + \sqrt{100}}

Q. Sum of the series

P = \frac{1}{2 \sqrt{1} + \sqrt{2}} + \frac{1}{3 \sqrt{2} + 2 \sqrt{3}} + . . . . + \frac{1}{100 \sqrt{99} + 99 \sqrt{100}}

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Evaluate: 1√2+1+1√3+√2+1√4+√3+...1√100+√99

The correct option is B 91√2+1=√2−1(√2+1)(√2−1)=√2−12−1=√2−1Similarly, 1√3+√2=√3−√2,1√4+√3=√4−√3,1√99+√98=√99−√98,1√100+√99=√100−√99∴ Given exp. =√2−1+√3−√2+√4−√3+....+√99−√98+√100−√99=√100−1=10−1=9.

Evaluate: $\frac{1}{\sqrt{2} + 1} + \frac{1}{\sqrt{3} + \sqrt{2}} + \frac{1}{\sqrt{4} + \sqrt{3}} + . . . \frac{1}{\sqrt{100} + \sqrt{99}}$