Can you generalize this?

Algebra Level 5

1 1 3 + 2 1 3 5 + 3 1 3 5 7 + \dfrac1{1\cdot3} + \dfrac2{1\cdot3\cdot5} + \dfrac3{1\cdot3\cdot5\cdot7} + \cdots

If the sum above can be expressed as A B \dfrac AB , where B B is a prime and A A is a positive integer, find ( A + B ) ! (A+B)! .


The answer is 6.

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1 solution

Rishabh Jain
Jan 16, 2016

1 2 ( 3 1 1 3 + 5 1 1 3 5 + 7 1 1 3 5 7 + ) \dfrac{1}{2}( \dfrac{3-1}{1\cdot3} + \dfrac{5-1}{1\cdot3\cdot5} + \dfrac{7-1}{1\cdot3\cdot5\cdot7} + \cdots ) = 1 2 ( ( 1 1 3 ) + ( 1 1 3 1 3 5 ) + ( 1 1 3 5 1 3 5 7 ) + ( 1 1 3 5 7 ) =\frac{1}{2}((1\color{#D61F06}{-\dfrac{1}{3})+(\dfrac{1}{1\cdot3}} \color{#3D99F6}{-\dfrac{1}{3\cdot5})+(\dfrac{1}{1\cdot3\cdot5}} \color{#20A900}{-\dfrac{1}{3\cdot5\cdot7})+(\dfrac{1}{1\cdot3\cdot5\cdot7}}\cdots) = 1 2 × 1 = 1 2 =\frac{1}{2}\times 1 =\frac{1}{2} ( 2 + 1 ) ! = 6 \Large\boxed{(2+1)!=6}

I really love ur 3 line solutions to summation problems! :)

Prakhar Bindal - 5 years, 5 months ago

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Thanks \Large\color{#D61F06}{\text{Thanks}}

Rishabh Jain - 5 years, 5 months ago

Wow! This is beautiful!

Pi Han Goh - 5 years, 4 months ago

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Thanks \Large\color{#69047E}{\text{Thanks}}

Rishabh Jain - 5 years, 4 months ago

Nice solution! Good work

Shreyash Rai - 5 years, 5 months ago

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You just presented a dreadful but elegant way of writing the simplest fraction :P

Swapnil Das - 5 years, 5 months ago

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Yeah XD making people lose a little confidence

Shreyash Rai - 5 years, 4 months ago

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@Shreyash Rai Took it from FIITJEE's worksheet, lol:-)😁

Department 8 - 5 years, 4 months ago

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@Department 8 Seemed like a fine question. Just curious what level would it be assigned

Shreyash Rai - 5 years, 4 months ago

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@Shreyash Rai OMG this is level 5 340 points!

Kaustubh Miglani - 5 years, 4 months ago

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@Kaustubh Miglani Ik right im shell shocked too

Shreyash Rai - 5 years, 4 months ago

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@Shreyash Rai Correction-400 points

Kaustubh Miglani - 5 years, 4 months ago

@Shreyash Rai This CPP question is viral!

Kushagra Yadav - 5 years, 4 months ago

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@Kushagra Yadav Its definetely rapidly gaining in popularity. That much is clear.

Shreyash Rai - 5 years, 4 months ago

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@Shreyash Rai Lvl 5 350!

Department 8 - 5 years, 4 months ago

@Shreyash Rai Whats ur score in JSTSE mock test 4 ?reply fast

Kaustubh Miglani - 5 years, 4 months ago

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@Kaustubh Miglani im getting much lesser this time than approximately everyone near me. all you need to know is its below 140.

Shreyash Rai - 5 years, 4 months ago

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@Shreyash Rai Is it even below 130?Btw what was ur rank in mock test 1 and 2 Please reply

Kaustubh Miglani - 5 years, 4 months ago

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@Kaustubh Miglani Not below 130 10th in test 1 pretty bad in test 2 but why do you want to know My marks? It should be of no significance to you

Shreyash Rai - 5 years, 4 months ago

@Kaustubh Miglani What is your score? I'm getting 154

Kushagra Yadav - 5 years, 4 months ago

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@Kushagra Yadav Its pretty much around ur score

Kaustubh Miglani - 5 years, 4 months ago

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@Kaustubh Miglani Tell me the number

Kushagra Yadav - 5 years, 4 months ago

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@Kushagra Yadav Well its a little low,155-157.I am NOT SURE.Think i will come 1st?

Kaustubh Miglani - 5 years, 4 months ago

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@Kaustubh Miglani That's obvious

Kushagra Yadav - 5 years, 4 months ago

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@Kushagra Yadav Nope it isnt,not at all.Well , I wanna be an expert in GK.Got some tips

Kaustubh Miglani - 5 years, 4 months ago

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@Kaustubh Miglani If all you guys are online check this .

Department 8 - 5 years, 4 months ago

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@Department 8 I wont solve it.Well let it gain level first .Hahahaha

Kaustubh Miglani - 5 years, 4 months ago

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@Kaustubh Miglani So what is your answer?

Department 8 - 5 years, 4 months ago

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@Department 8 That was an easy one.

Shreyash Rai - 5 years, 4 months ago

@Kaustubh Miglani are u sure u want me to post answer here?

Kaustubh Miglani - 5 years, 4 months ago

@Kushagra Yadav Ur score in real JSTSE is too good.My score is 174-175(not sure of 1 question) We will surely get close ranks.BTW,whats ur score in GK and SCience and maths individually?

Kaustubh Miglani - 5 years, 4 months ago

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@Kaustubh Miglani Damn why dont you do silly mistakes? How much dd you get in gk by the way?

Shreyash Rai - 5 years, 4 months ago

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@Shreyash Rai I do .I could have got 175-176 Tell ur marks and I will mine

Kaustubh Miglani - 5 years, 4 months ago

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@Kaustubh Miglani forget my marks. the official key is out by the way. you could check your marks nicely the link is http://www.edudel.nic.in/upload 2015 16/1782 dt 02022016.pdf

Shreyash Rai - 5 years, 4 months ago

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@Shreyash Rai Can u please send me the answer key ?Its not opening .Please Here is my email id nowiamachangedman@ gmail.com

Kaustubh Miglani - 5 years, 4 months ago

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@Kaustubh Miglani too lazy bro. well if you cant directly open the link then go to edudel.nic.in. then there would be an option called public circulars. under that you would see the ans key. if the site only doesnt open..........have fun

Shreyash Rai - 5 years, 4 months ago

@Kaustubh Miglani gk 33 phy 36 chem, bio 39 maths 27 what abut you??

Kushagra Yadav - 5 years, 4 months ago

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@Kushagra Yadav gk 32 or 33 phy-34 sob sob sob chem-38 bio-40 maths-30 Think I CAN COME 1ST in delhi?

Kaustubh Miglani - 5 years, 4 months ago

@Department 8 Which phase?

Priyanshu Mishra - 5 years, 4 months ago

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@Priyanshu Mishra Our review classes are going on.

Department 8 - 5 years, 4 months ago

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@Department 8 Wouldnt think that this ordinary cpp question would become lvl 5. Interesting it is.

Shreyash Rai - 5 years, 4 months ago

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@Shreyash Rai How many of your class could answer?

Swapnil Das - 5 years, 4 months ago

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@Swapnil Das Actually I was the only one who answered. But then 2 or 3 Like lakshay I beleive May not have tried it

Shreyash Rai - 5 years, 4 months ago

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@Shreyash Rai My goodness, it's 335 points now.

Swapnil Das - 5 years, 4 months ago

Thanks \Large\color{#20A900}{\text{Thanks}}

Rishabh Jain - 5 years, 4 months ago

I'm amazed !!

Akshat Sharda - 5 years, 4 months ago

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😋 \Huge😋

Rishabh Jain - 5 years, 4 months ago

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You live in ajmer ? I do !!

Akshat Sharda - 5 years, 4 months ago

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@Akshat Sharda You also ?? But your profile on Brilliant shows you are from Chandigarh...

Rishabh Jain - 5 years, 4 months ago

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@Rishabh Jain Yeah !! It is wrong !! I live in Ajmer and I'm in Mayoor School.

Akshat Sharda - 5 years, 4 months ago

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@Akshat Sharda 10 class ??

Rishabh Jain - 5 years, 4 months ago

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@Rishabh Jain Yes ! What about you ?

Akshat Sharda - 5 years, 4 months ago

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@Akshat Sharda I'm in 12th .

Rishabh Jain - 5 years, 4 months ago

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@Rishabh Jain You guys are in the same school?

Department 8 - 5 years, 4 months ago

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@Department 8 Nope... Just the same city..... Ajmer :)

Rishabh Jain - 5 years, 4 months ago

It has became HIGHLY over-rated . hardly a 150 point problem got a rating of 350 OMG!

Prakhar Bindal - 5 years, 4 months ago

400 points FIITJEE CPP problem! Its overrated man!

Swapnil Das - 5 years, 4 months ago

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i expected the question to be at lvl 3 or 4 at most. i was sooo wrong.

Shreyash Rai - 5 years, 4 months ago

Lovely telescoping!

Mahdi Raza - 11 months, 2 weeks ago

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