1 - 1- 2 - 1 - 2 - 3 - 1 - 3 - 4 - 1 - 15 - 16 -

The correct option is D 3 T_n amp;=1√(n)+√(n+1)=√(n) √(n+1)(√(n)+√(n+1))(√(n) √(n+1)) amp;=√(n) √(n+1)(n) (n+1)=√(n+1) √(n) ∑_n=1^15 T_n= a ...

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Byju's Answer

Standard XI

Mathematics

Law of Reciprocal

1 / 1+√ 2 + 1...

Question

$\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + . . . . . . + \frac{1}{\sqrt{15} + \sqrt{16}} =_____$

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Solution

The correct option is D 3
$\begin{matrix} T_{n} & = \frac{1}{\sqrt{n} + \sqrt{n + 1}} = \frac{\sqrt{n} - \sqrt{n + 1}}{(\sqrt{n} + \sqrt{n + 1}) (\sqrt{n} - \sqrt{n + 1})} = \frac{\sqrt{n} - \sqrt{n + 1}}{(n) - (n + 1)} = \sqrt{n + 1} - \sqrt{n} \end{matrix}$
$\begin{matrix} 15 \sum n = 1 T_{n} = & \sqrt{2} - \sqrt{1} + \sqrt{3} - \sqrt{2} + \sqrt{4} - \sqrt{3} \cdot \dots + \sqrt{16} - \sqrt{15} = & \sqrt{16} - \sqrt{1} = 4 - 1 = 3 \end{matrix}$
Answer: option $(D)$

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Similar questions

Q. The value of

[\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{16} + \sqrt{15}}]

is equal to

1+1/2+1/3+...+1/1023

Prove that :

$(i) (\sqrt{3 \times 5^{- 3}} \div \sqrt[3]{3^{- 1}} \sqrt{5}) \times \sqrt[6]{3 \times 5^{6}} = \frac{3}{5} (i i) 9^{\frac{3}{2}} - 3 \times 5^{0} - {(\frac{1}{81})}^{\frac{- 1}{2}} = 15 (i i i) {(\frac{1}{4})}^{- 2} - 3 \times 8^{\frac{2}{3}} \times 4^{0} + {(\frac{9}{16})}^{\frac{- 1}{2}} = \frac{16}{3} (i v) \frac{2^{\frac{1}{2}} \times 3^{\frac{1}{3}} \times 4^{\frac{1}{4}}}{10^{\frac{- 1}{5}} \times 5^{\frac{3}{5}}} \div \frac{3^{\frac{4}{3}} \times 5^{\frac{- 7}{5}}}{4^{\frac{- 3}{5}} \times 6} = 10 (v) \sqrt{\frac{1}{4}} + (0.01)^{\frac{- 1}{2}} - (27)^{\frac{2}{3}} = \frac{3}{2} (v i) \frac{2^{n} + 2^{n - 1}}{2^{n + 1} - 2^{n}} = \frac{3}{2} (v i i) {(\frac{64}{125})}^{\frac{- 2}{3}} + \frac{1}{{(\frac{256}{625})}^{\frac{1}{4}}} + {(\frac{\sqrt{25}}{\sqrt[3]{64}})}^{0} = \frac{61}{16} (v i i i) \frac{3^{- 3} \times 6^{2} \times \sqrt{98}}{5^{2} \times \sqrt[3]{\frac{1}{25}} \times (15)^{\frac{- 4}{3}} \times 3^{\frac{1}{3}}} = 28 \sqrt{2} (i x) \frac{(0.6)^{0} - (0.1)^{- 1}}{{(\frac{3}{8})}^{- 1} {(\frac{3}{2})}^{3} + {(\frac{1}{3})}^{- 1}} = - \frac{3}{2}$

Q. Calculate:

1 − 2 + 3 − 4 + 5 − 6 +......+ 15 − 16

\int {\sqrt{2 x^{2} - 3 x + 1}}_{d x}

will be equal to -

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11+√2+1√2+√3+1√3+√4+......+1√15+√16=_____

The correct option is D 3Tn=1√n+√n+1=√n−√n+1(√n+√n+1)(√n−√n+1)=√n−√n+1(n)−(n+1)=√n+1−√n15∑n=1Tn=√2−√1+√3−√2+√4−√3⋅⋯+√16−√15=√16−√1=4−1=3Answer: option (D)

$\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + . . . . . . + \frac{1}{\sqrt{15} + \sqrt{16}} =_____$