Find-the-sum-9-1-2-10-2-2-3-10-3-3-4-10-99-99-100-x-the-greatest-integer-function-

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

F i n d t h e s u m

\frac{9}{1 + \sqrt{2}} + \frac{10 - ⌊ \sqrt{2} ⌋}{\sqrt{2} + \sqrt{3}} + \frac{10 - ⌊ \sqrt{3} ⌋}{\sqrt{3} + \sqrt{4}} + \dots + \frac{10 - ⌊ \sqrt{99} ⌋}{\sqrt{99} + \sqrt{100}}

⌊ x ⌋ = t h e g r e a t e s t i n t e g e r f u n c t i o n

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

⌊ \sqrt{1} ⌋ to ⌊ \sqrt{3} ⌋ ∽ 1

⌊ \sqrt{4} ⌋ to ⌊ \sqrt{8} ⌋ ∽ 2

⌊ \sqrt{9} ⌋ to ⌊ \sqrt{15} ⌋ ∽ 3

:

:

⌊ \sqrt{81} ⌋ to ⌊ \sqrt{99} ⌋ ∽ 9

∴ the series becomes

9 (\frac{1}{\sqrt{1} + \sqrt{2}} + . . + \frac{1}{\sqrt{3} + \sqrt{4}}) + 8 (\frac{1}{\sqrt{4} + \sqrt{5}} + . . + \frac{1}{\sqrt{8} + \sqrt{9}})

+ 7 (\frac{1}{\sqrt{9} + \sqrt{10}} + . . + \frac{1}{\sqrt{15} + \sqrt{16}}) + \dots \dots 2 (\frac{1}{\sqrt{64} + \sqrt{65}} + . .

\frac{1}{\sqrt{80} + \sqrt{81}}) + 1 (\frac{1}{\sqrt{81} + \sqrt{82}} + . . + \frac{1}{\sqrt{99} + \sqrt{100}})

= 9 (\sqrt{4} - 1) + 8 (\sqrt{9} - \sqrt{4}) + 7 (\sqrt{16} - \sqrt{9}) + \dots . + 2 (\sqrt{81} - \sqrt{64})

+ 1 (\sqrt{100} - \sqrt{81})

= 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1

= 45 please check .

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

35 + 3 \sqrt{11}

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

n i c e s o l u t i o n!

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

am I wrong Sir ?

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

thank you sir! I fix it .

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

Thanks Sir!

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

t h e a n s w e r i s 135

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

i f t h e a n s w e r i s 135, t h e n y o u r q u e s t i o n

i s w r o n g . p l e a s e r e c h e c k!

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

I t h i n k t h e b o o k i s w r o n g .

I w i l l c h e c k t h e o r i g i n a l q u e s t i o n .