Prove thati 13-7-17-5-15-3-13-1-1ii 11-2-12-3-13-4-14-5-15-6-16-7-17-8-18-9-2

(i) 13+7+17+5+15+3+13+1=13+7 #215;3 73 7+17+5 #215;7 57 5+15+3 #215;5 35 3+13+1 #215;3 13 1=3 732 72+7 572 52+5 352 32+3 132 12 =3 79 7+7 57 5+5 35 3+3 ...

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Byju's Answer

Standard VI

Mathematics

Organised Data

Prove thati 1...

Question

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

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Solution

(i)
To prove: $\frac{1}{3 + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{3}} + \frac{1}{\sqrt{3} + 1} = 1$
Lets take LHS and then equate it to RHS.
LHS $= \frac{1}{3 + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{3}} + \frac{1}{\sqrt{3} + 1}$
Lets do rationalize the denominator of each term,
$= [\frac{1}{3 + \sqrt{7}} \times \frac{3 - \sqrt{7}}{3 - \sqrt{7}}] + [\frac{1}{\sqrt{7} + \sqrt{5}} \times \frac{\sqrt{7} - \sqrt{5}}{\sqrt{7} - \sqrt{5}}] + [\frac{1}{\sqrt{5} + \sqrt{3}} \times \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}}] + [\frac{1}{\sqrt{3} + 1} \times \frac{\sqrt{3} - 1}{\sqrt{3} - 1}]$
$= \frac{3 - \sqrt{7}}{3^{2} - (\sqrt{7})^{2}} + \frac{\sqrt{7} - \sqrt{5}}{(\sqrt{7})^{2} - (\sqrt{5})^{2}} + \frac{\sqrt{5} - \sqrt{3}}{(\sqrt{5})^{2} - (\sqrt{3})^{2}} + \frac{\sqrt{3} - 1}{(\sqrt{3})^{2} - 1^{2}}$
$= \frac{3 - \sqrt{7}}{9 - 7} + \frac{\sqrt{7} - \sqrt{5}}{7 - 5} + \frac{\sqrt{5} - \sqrt{3}}{5 - 3} + \frac{\sqrt{3} - 1}{3 - 1}$
$= \frac{3 - \sqrt{7}}{2} + \frac{\sqrt{7} - \sqrt{5}}{2} + \frac{\sqrt{5} - \sqrt{3}}{2} + \frac{\sqrt{3} - 1}{2}$
$= \frac{3 - \sqrt{7} + \sqrt{7} - \sqrt{5} + \sqrt{5} - \sqrt{3} + \sqrt{3} - 1}{2}$
$= \frac{2}{2}$
$= 1$
$=$ RHS
$∴$ LHS $=$ RHS
Hence, proved.
(ii)
To prove:
$\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{4} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{8}} + \frac{1}{\sqrt{8} + \sqrt{9}} = 2$
Lets take LHS and then equate it to RHS.
LHS $= \frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{4} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{8}} + \frac{1}{\sqrt{8} + \sqrt{9}}$
Lets do rationalize the denominator of each term,
$= [\frac{1}{1 + \sqrt{2}} \times \frac{\sqrt{2} - 1}{\sqrt{2} - 1}] + [\frac{1}{\sqrt{2} + \sqrt{3}} \times \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}}] + [\frac{1}{\sqrt{4} + \sqrt{3}} \times \frac{\sqrt{4} - \sqrt{3}}{\sqrt{4} - \sqrt{3}}] + [\frac{1}{\sqrt{4} + \sqrt{5}} \times \frac{\sqrt{5} - \sqrt{4}}{\sqrt{5} - \sqrt{4}}] + [\frac{1}{\sqrt{5} + \sqrt{6}} \times \frac{\sqrt{6} - \sqrt{5}}{\sqrt{6} - \sqrt{5}}] + [\frac{1}{\sqrt{6} + \sqrt{7}} \times \frac{\sqrt{7} - \sqrt{6}}{\sqrt{7} - \sqrt{6}}] + [\frac{1}{\sqrt{7} + \sqrt{8}} \times \frac{\sqrt{8} - \sqrt{7}}{\sqrt{8} - \sqrt{7}}] + [\frac{1}{\sqrt{8} + \sqrt{9}} \times \frac{\sqrt{9} - \sqrt{8}}{\sqrt{9} - \sqrt{8}}]$
$= \frac{\sqrt{2} - 1}{(\sqrt{2})^{2} - 1^{2}} + \frac{\sqrt{3} - \sqrt{2}}{(\sqrt{3})^{2} - (\sqrt{2})^{2}} + \frac{\sqrt{4} - \sqrt{3}}{(\sqrt{4})^{2} - (\sqrt{3})^{2}} + \frac{\sqrt{5} - \sqrt{4}}{(\sqrt{5})^{2} - (\sqrt{4})^{2}} + \frac{\sqrt{6} - \sqrt{5}}{(\sqrt{6})^{2} - (\sqrt{5})^{2}} + \frac{\sqrt{7} - \sqrt{6}}{(\sqrt{7})^{2} - (\sqrt{6})^{2}} + \frac{\sqrt{8} - \sqrt{7}}{(\sqrt{8})^{2} - (\sqrt{7})^{2}} + \frac{\sqrt{9} - \sqrt{8}}{(\sqrt{9})^{2} - (\sqrt{8})^{2}}$
$= \frac{\sqrt{2} - 1}{2 - 1} + \frac{\sqrt{3} - \sqrt{2}}{3 - 2} + \frac{\sqrt{4} - \sqrt{3}}{4 - 3} + \frac{\sqrt{5} - \sqrt{4}}{5 - 4} + \frac{\sqrt{6} - \sqrt{5}}{6 - 5} + \frac{\sqrt{7} - \sqrt{6}}{7 - 6} + \frac{\sqrt{8} - \sqrt{7}}{8 - 7} + \frac{\sqrt{9} - \sqrt{8}}{9 - 8}$
$= \frac{\sqrt{2} - 1}{1} + \frac{\sqrt{3} - \sqrt{2}}{1} + \frac{\sqrt{4} - \sqrt{3}}{1} + \frac{\sqrt{5} - \sqrt{4}}{1} + \frac{\sqrt{6} - \sqrt{5}}{1} + \frac{\sqrt{7} - \sqrt{6}}{1} + \frac{\sqrt{8} - \sqrt{7}}{1} + \frac{\sqrt{9} - \sqrt{8}}{1}$
$= \sqrt{2} - 1 + \sqrt{3} - \sqrt{2} + \sqrt{4} - \sqrt{3} + \sqrt{5} - \sqrt{4} + \sqrt{6} - \sqrt{5} + \sqrt{7} - \sqrt{6} + \sqrt{8} - \sqrt{7} + \sqrt{9} - \sqrt{8}$
$= \sqrt{9} - 1$
$= 3 - 1$
$= 2$
$∴$ LHS $=$ RHS
So, $\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{4} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{8}} + \frac{1}{\sqrt{8} + \sqrt{9}} = 2$
Hence, proved.

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Similar questions

Re-arrange suitably and find the sum in each of the following :

(i) $\frac{11}{12} + \frac{- 17}{3} + \frac{11}{2} + \frac{- 25}{2}$

(ii) $\frac{- 6}{7} + \frac{- 5}{6} + \frac{- 4}{9} + \frac{- 15}{7}$

(iii) $\frac{3}{5} + \frac{7}{3} + \frac{9}{5} + \frac{- 13}{15} + \frac{- 7}{3}$

(iv) $\frac{4}{13} + \frac{- 5}{8} + \frac{- 8}{13} + \frac{9}{13}$

(v) $\frac{2}{3} + \frac{- 4}{5} + \frac{1}{3} + \frac{2}{5}$

(vi) $\frac{1}{8} + \frac{5}{12} + \frac{2}{7} + \frac{7}{12} + \frac{9}{7} + \frac{- 5}{16}$

Verify the following :

(i) $\frac{3}{7} \times (\frac{5}{6} + \frac{12}{13}) = (\frac{3}{7} \times \frac{5}{6}) + (\frac{3}{7} \times \frac{12}{13})$
(ii) $\frac{- 15}{4} \times (\frac{3}{7} + \frac{- 12}{5}) = (\frac{- 15}{4} \times \frac{3}{7}) + (\frac{- 15}{4} \times \frac{- 12}{5})$
(iii) $(\frac{- 8}{3} + \frac{- 13}{12}) \times \frac{5}{6} = (\frac{- 8}{3} \times \frac{5}{6}) + (\frac{- 13}{12} \times \frac{5}{6})$
(iv) $\frac{- 16}{7} \times (\frac{- 8}{9} + \frac{- 7}{6}) = (\frac{- 16}{7} \times \frac{- 8}{9}) + (\frac{- 16}{7} \times \frac{- 7}{6})$

Q. The distance (in km) of

40

engineers from their residence to their place of work were found as follows:

5, 3, 10, 20, 25, 11, 13, 7, 12, 31, 19, 10, 12, 17, 18, 11, 32, 17, 16, 2, 7, 9, 7, 8,

3, 5, 12, 15, 18, 3, 12, 14, 2, 9, 6, 15, 15, 7, 6, 12

How many engineers travelled distance more than

14

and less than

26

Q. Show that

\frac{1}{(3 - \sqrt{8})} - \frac{1}{(\sqrt{8} - \sqrt{7})} + \frac{1}{(\sqrt{7} - \sqrt{6})} - \frac{1}{(\sqrt{6} - \sqrt{5})} + \frac{1}{(\sqrt{5} - 2)} = 5

Q. Without adding, find the sum:
(i) (1 + 3 + 5 + 7 + 9 + 11 + 13)
(ii) (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19)
(iii) (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21 + 23)

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Prove that:(i) 13+√7+1√7+√5+1√5+√3+1√3+1=1(ii) 11+√2+1√2+√3+1√3+√4+1√4+√5+1√5+√6+1√6+√7+1√7+√8+1√8+√9=2