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Question Number 91328 by Maclaurin Stickker last updated on 30/Apr/20

Find the sum  (9/(1+(√2)))+((10−⌊(√2)⌋)/((√2)+(√3)))+((10−⌊(√3)⌋)/((√3)+(√4)))+...+((10−⌊(√(99))⌋)/((√(99))+(√(100))))  ⌊x⌋=the greatest integer function

$${Find}\:{the}\:{sum} \\ $$$$\frac{\mathrm{9}}{\mathrm{1}+\sqrt{\mathrm{2}}}+\frac{\mathrm{10}−\lfloor\sqrt{\mathrm{2}}\rfloor}{\sqrt{\mathrm{2}}+\sqrt{\mathrm{3}}}+\frac{\mathrm{10}−\lfloor\sqrt{\mathrm{3}}\rfloor}{\sqrt{\mathrm{3}}+\sqrt{\mathrm{4}}}+...+\frac{\mathrm{10}−\lfloor\sqrt{\mathrm{99}}\rfloor}{\sqrt{\mathrm{99}}+\sqrt{\mathrm{100}}} \\ $$$$\lfloor{x}\rfloor={the}\:{greatest}\:{integer}\:{function} \\ $$

Commented by Prithwish Sen 1 last updated on 30/Apr/20

⌊(√1)⌋ to ⌊(√3)⌋   ∽  1   ⌊(√4)⌋  to ⌊(√8)⌋  ∽  2  ⌊(√9)⌋ to  ⌊(√(15))⌋∽ 3              :              :   ⌊(√(81))⌋ to ⌊(√(99))⌋ ∽ 9  ∴ the series becomes  9((1/((√1)+(√2))) +..+(1/((√3)+(√4))))+8((1/((√4)+(√5)))+..+(1/((√8)+(√9))))  +7((1/((√9)+(√(10))))+..+(1/((√(15))+(√(16))))) +......2((1/((√(64))+(√(65))))+..  (1/((√(80))+(√(81)))))+1((1/((√(81))+(√(82)))) +..+(1/((√(99))+(√(100)))))  =9((√4)−1)+8((√9)−(√4))+7((√(16))−(√9))+....+2((√(81))−(√(64)))  +1((√(100))−(√(81)))  =9+8+7+6+5+4+3+2+1  =45   please check.

$$\lfloor\sqrt{\mathrm{1}}\rfloor\:\mathrm{to}\:\lfloor\sqrt{\mathrm{3}}\rfloor\:\:\:\backsim\:\:\mathrm{1}\: \\ $$$$\lfloor\sqrt{\mathrm{4}}\rfloor\:\:\mathrm{to}\:\lfloor\sqrt{\mathrm{8}}\rfloor\:\:\backsim\:\:\mathrm{2} \\ $$$$\lfloor\sqrt{\mathrm{9}}\rfloor\:\mathrm{to}\:\:\lfloor\sqrt{\mathrm{15}}\rfloor\backsim\:\mathrm{3} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\::\: \\ $$$$\lfloor\sqrt{\mathrm{81}}\rfloor\:\mathrm{to}\:\lfloor\sqrt{\mathrm{99}}\rfloor\:\backsim\:\mathrm{9} \\ $$$$\therefore\:\mathrm{the}\:\mathrm{series}\:\mathrm{becomes} \\ $$$$\mathrm{9}\left(\frac{\mathrm{1}}{\sqrt{\mathrm{1}}+\sqrt{\mathrm{2}}}\:+..+\frac{\mathrm{1}}{\sqrt{\mathrm{3}}+\sqrt{\mathrm{4}}}\right)+\mathrm{8}\left(\frac{\mathrm{1}}{\sqrt{\mathrm{4}}+\sqrt{\mathrm{5}}}+..+\frac{\mathrm{1}}{\sqrt{\mathrm{8}}+\sqrt{\mathrm{9}}}\right) \\ $$$$+\mathrm{7}\left(\frac{\mathrm{1}}{\sqrt{\mathrm{9}}+\sqrt{\mathrm{10}}}+..+\frac{\mathrm{1}}{\sqrt{\mathrm{15}}+\sqrt{\mathrm{16}}}\right)\:+......\mathrm{2}\left(\frac{\mathrm{1}}{\sqrt{\mathrm{64}}+\sqrt{\mathrm{65}}}+..\right. \\ $$$$\left.\frac{\mathrm{1}}{\sqrt{\mathrm{80}}+\sqrt{\mathrm{81}}}\right)+\mathrm{1}\left(\frac{\mathrm{1}}{\sqrt{\mathrm{81}}+\sqrt{\mathrm{82}}}\:+..+\frac{\mathrm{1}}{\sqrt{\mathrm{99}}+\sqrt{\mathrm{100}}}\right) \\ $$$$=\mathrm{9}\left(\sqrt{\mathrm{4}}−\mathrm{1}\right)+\mathrm{8}\left(\sqrt{\mathrm{9}}−\sqrt{\mathrm{4}}\right)+\mathrm{7}\left(\sqrt{\mathrm{16}}−\sqrt{\mathrm{9}}\right)+....+\mathrm{2}\left(\sqrt{\mathrm{81}}−\sqrt{\mathrm{64}}\right) \\ $$$$+\mathrm{1}\left(\sqrt{\mathrm{100}}−\sqrt{\mathrm{81}}\right) \\ $$$$=\mathrm{9}+\mathrm{8}+\mathrm{7}+\mathrm{6}+\mathrm{5}+\mathrm{4}+\mathrm{3}+\mathrm{2}+\mathrm{1} \\ $$$$=\mathrm{45}\:\:\:\boldsymbol{\mathrm{please}}\:\boldsymbol{\mathrm{check}}. \\ $$

Commented by Prithwish Sen 1 last updated on 30/Apr/20

35+3(√(11))

$$\mathrm{35}+\mathrm{3}\sqrt{\mathrm{11}} \\ $$

Commented by mr W last updated on 30/Apr/20

nice solution!

$${nice}\:{solution}! \\ $$

Commented by Prithwish Sen 1 last updated on 30/Apr/20

am I wrong Sir ?

$$\mathrm{am}\:\mathrm{I}\:\mathrm{wrong}\:\mathrm{Sir}\:? \\ $$

Commented by Prithwish Sen 1 last updated on 30/Apr/20

thank you sir ! I fix it.

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir}\:!\:\mathrm{I}\:\mathrm{fix}\:\mathrm{it}. \\ $$

Commented by Prithwish Sen 1 last updated on 30/Apr/20

Thanks Sir !

$$\mathrm{Thanks}\:\mathrm{Sir}\:! \\ $$

Commented by Maclaurin Stickker last updated on 30/Apr/20

the answer is 135

$${the}\:{answer}\:{is}\:\mathrm{135} \\ $$

Commented by mr W last updated on 30/Apr/20

if the answer is 135, then your question  is wrong. please recheck!

$${if}\:{the}\:{answer}\:{is}\:\mathrm{135},\:{then}\:{your}\:{question} \\ $$$${is}\:{wrong}.\:{please}\:{recheck}! \\ $$

Commented by Maclaurin Stickker last updated on 30/Apr/20

I think the book is wrong.  I will check the original question.

$${I}\:{think}\:{the}\:{book}\:{is}\:{wrong}. \\ $$$${I}\:{will}\:{check}\:{the}\:{original}\:{question}. \\ $$

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