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No.name
 11 months ago
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No.name
 11 months ago
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No.name
 11 months ago
Best ResponseYou've already chosen the best response.2If \[a _{i}>0 \forall i \in N\] such that \[\prod_{n}^{i=1}a _{i}=1\] Then prove that \[(1+a _{1})(1+a _{2})(1+a _{3})(1+a _{4}).....(1+a _{n}) \ge 2^{n}\]

No.name
 11 months ago
Best ResponseYou've already chosen the best response.2@ikram002p @ganeshie8

No.name
 11 months ago
Best ResponseYou've already chosen the best response.2What does the sign inverted u mean

ikram002p
 11 months ago
Best ResponseYou've already chosen the best response.0ive seen this before :O

ganeshie8
 11 months ago
Best ResponseYou've already chosen the best response.2its like \(\sum \), \[\large \prod_{k=1}^{3}k= (1)(2)(3)\]

ganeshie8
 11 months ago
Best ResponseYou've already chosen the best response.2\[\large \prod_{k=1}^{3}k^2= (1^2)(2^2)(3^2)\]

ganeshie8
 11 months ago
Best ResponseYou've already chosen the best response.2the proof for the original question is simple

No.name
 11 months ago
Best ResponseYou've already chosen the best response.2Yes then it is easy after i understood what the sign means

No.name
 11 months ago
Best ResponseYou've already chosen the best response.2i will be back after dinner

ganeshie8
 11 months ago
Best ResponseYou've already chosen the best response.2yes :) here is an one line proof : since \(a_i \in \mathbb{N}\), \(a_i+1 \ge 2 \implies \prod \limits_{i=1}^{n}(a _{i}+1)\ge 2^n \)

ikram002p
 11 months ago
Best ResponseYou've already chosen the best response.0mmm sure its a proof ?

ikram002p
 11 months ago
Best ResponseYou've already chosen the best response.0i thought i had to do this \(\prod_{n}^{i=1}a _{i}\prod_{n}^{i=1} (1+\dfrac{1}{a_i}) \) but i got confused for a second sense\( a_i \) is not integer check the product =1

ikram002p
 11 months ago
Best ResponseYou've already chosen the best response.0so \(0<a_i\le 1\) do u agree to this @ganeshie8 ?

ikram002p
 11 months ago
Best ResponseYou've already chosen the best response.0wait it can be rational but not integer

ikram002p
 11 months ago
Best ResponseYou've already chosen the best response.0forget about the interval

ganeshie8
 11 months ago
Best ResponseYou've already chosen the best response.2\[(1+a _{1})(1+a _{2})(1+a _{3})(1+a _{4}) \cdots (1+a _{n}) = \prod \limits_{i=1}^{n} (a_i + 1)\]

ganeshie8
 11 months ago
Best ResponseYou've already chosen the best response.2ahh i see your point, so we're given that \(a_i \gt 0\), we're NOT given that its a natural number

ikram002p
 11 months ago
Best ResponseYou've already chosen the best response.0yep exactly and sence product of a_i = 1 then for sure it cant be integers but also it can be <1 or >1

ganeshie8
 11 months ago
Best ResponseYou've already chosen the best response.2okay then it cannot be an one liner >.<

ikram002p
 11 months ago
Best ResponseYou've already chosen the best response.0\(a_i \) should be rationl even though its confusing in that case what if \(a_i \le \) 1 then problem solved but if \(a_i >1 \)like 3/2 or 5/2 then we can do the first line proof u have but what if \(a_i \) was mixed btw number which >1 or <1 or =1 then we can't conclude anything in this case

ganeshie8
 11 months ago
Best ResponseYou've already chosen the best response.2lets expand the product

ikram002p
 11 months ago
Best ResponseYou've already chosen the best response.0the thing is im too hungry lol xD so quick i wanna sleep hehe unless there is a given condition for \(a_i\)

ikram002p
 11 months ago
Best ResponseYou've already chosen the best response.0see ganesh for example (1+1/2) (1+1/3) (1+1/4) <2^3 (1+1/3776568)(1+3/2)(1+1/2)(1/45456) <2^4 ( not sure :P)

ikram002p
 11 months ago
Best ResponseYou've already chosen the best response.0only this case work , if \(1\le a_i \) (1+3/2) (1+5/4) (1+7/6) >=2^3

ikram002p
 11 months ago
Best ResponseYou've already chosen the best response.0nope we cant , we only can conclude that \(a_i\) is rational

ikram002p
 11 months ago
Best ResponseYou've already chosen the best response.0for example \(\dfrac{1}{2}×\dfrac{2}{3}×\dfrac{3}{1}×\dfrac{1}{1}=1\)

ikram002p
 11 months ago
Best ResponseYou've already chosen the best response.0gtg nw , have a nice day

No.name
 11 months ago
Best ResponseYou've already chosen the best response.2dw:1404314484947:dw \[(1+a _{1})(1+a _{2})(1+a _{3}).....(1+a _{n})\ge2^{n}[(a _{1}a _{2}a _{3}..a _{n}]^{1/2}\] dw:1404314863576:dw so proved

ganeshie8
 11 months ago
Best ResponseYou've already chosen the best response.2nice :) AM/GM inequality to the rescue !!

No.name
 11 months ago
Best ResponseYou've already chosen the best response.2yes, just didn't knew what that sign meant

ikram002p
 11 months ago
Best ResponseYou've already chosen the best response.0cool O.O never seen this before

ganeshie8
 11 months ago
Best ResponseYou've already chosen the best response.2AMGM inequality : \(\dfrac{a+b}{2} \ge \sqrt{ab}\)

ikram002p
 11 months ago
Best ResponseYou've already chosen the best response.0ic .. i dont think that ive seen it before :D
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