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shubhamsrg
 2 years ago
how to prove
n! ≤ ( (n+1)/2 )^n
for n≥ 1
shubhamsrg
 2 years ago
how to prove n! ≤ ( (n+1)/2 )^n for n≥ 1

This Question is Closed

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.1Maybe I'm wrong, but I was under the impression that factorials are defined only for \([0,\infty)\). So isn't it just a case of demonstrating the conjecture for \(0,1\)? I barely recall reading a paper about evaluating negative factorials with the gamma function, but that was entirely on the complex plane (IIRC).

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.2ohh lol..sorry..it should be >= sign,,wait i'll edit the ques..

badreferences
 2 years ago
Best ResponseYou've already chosen the best response.1Lemme dust off my real analysis textbook... ahaha...

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1n! = 1*2*3*...*(n2)*(n1)*n Now, (1+n/2) = (2+n1)/2 = (3+n2)/2

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1(1+n/2) = (2+n1)/2 = (3+n2)/2 ==== (1+n)/2 Also, n! has n number of terms..... so there will be around half number of (1+n)/2 terms..... and also, ( (n+1)/2 )^n has n number of terms.. Thus, n! ≤ ( (n+1)/2 )^n

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1I hope I made it clear

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.2sorry didnt get it :

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1where didnt u understand

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1n! = 1*2*3*...*(n2)*(n1)*n

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1right? @shubhamsrg

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.2so there will be around half number of (1+n)/2 terms..... + conclusion part..

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1Now, take 1 and n and average them

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1So, u will get (n+1)/2 right?

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1u will again get (n+1)/2 right?

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.2maybe you meant 2 and n1 ?

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1oh...... 2 and (n1)

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.2okay,,i get that part..

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.2check saura's method and also this go by mathematical induction... Use this :\[2\le (\frac{n+2}{n+1})^{n+1}\]

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1guys I think I was totally wrong

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.1But got the method to solve this

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.2@mukushla didnt get you.. @sauravshakya your method ?

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.2here's what i tried to do.. (n+1)/2 = n/2 + 1/2 now for even n, n/2 + 1 > (n+1)/2 > n/2 (n/2 + 1) (n/2 + 2) > ((n+1)/2)((n+1)/2) > (n/2) (n/2 1) . . . . (n/2 +1)(n/2 +2)......(n/2 + n/2) > ((n+1)/2 )^(n/2) > (n/2) (n/2 1).....(n/2 n/2) n! / (n/2) ! > ((n+1)/2 )^(n/2) > (n/2) ! is this any help? how can we take it further ?

ravimeena
 2 years ago
Best ResponseYou've already chosen the best response.0take n=2 so n!=2 and (3/2)^2=2.25 so 2<2.25 hence proved

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.2well thats ofcorse not a general soln..

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.2u got n! / (n/2) ! > ((n+1)/2 )^(n/2) how can we match this with original inequality?

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.2you're the expert,,you telll!! :P

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.2lol...idk if it will help or not.

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.2what were you saying about induction?

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.2well for \(n=1\) its true... let suppose\[n! \le (\frac{n+1}{2})^n \]be a true statement...we just need to prove\[(n+1)! \le (\frac{n+2}{2})^{n+1} \]

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.2\[(n+1)!=(n+1).n! \le (n+1).(\frac{n+1}{2})^{n}\le \frac{n+1}{2}(\frac{n+2}{n+1})^{n+1}.(\frac{n+1}{2})^{n}=(\frac{n+2}{2})^{n+1} \]

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.2didnt get the second last step..?

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.2well \[1 \le \frac{1}{2} (\frac{n+2}{n+1})^{n+1}\] but u need to prove this inequality also

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.2if we try like this : n = (n+1 /2) + (n1 /2) n1 = (n+1 /2) + (n3 /2) . . . . and multiplying all, n! = (n+1 /2)^n + something.. does this help ??

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.2RHS will be some polynomial or f(n+1 /2) with roots (n1)/2 ,(n3)/2 etc.. so there isnt much problem multiplying all..main problem is how to prove!! ?? !!

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.2u most prove \(\text{something} \le 0\) right?

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.2if n+1 /2 = x, then it'll be of the form x^n + x^(n1) (0)  x^(n2)(something +ve) + ............... what will come in ............ ?? am quite sure about what i wrote before ...............

satellite73
 2 years ago
Best ResponseYou've already chosen the best response.0i did not read all of the above, but is this supposed to be a proof by induction?

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.2well you can use any legitimate method sir..

satellite73
 2 years ago
Best ResponseYou've already chosen the best response.0no i messed that up, have to try again

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.2people i got the solution!!

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.2it was accidently,,while solving another ques,,and i recalled AM >= GM,,here's my solution

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.2we know that : [ (1 + 2 +3 ..... n)/n ] ^n >= 1.2.3......n or [ n(n+1)/2n ]^n >= n! => (n+1 /2)^n >=n! lol...

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.2very nice man...\[\frac{1+2+...+n}{n}\ge\sqrt[n]{n!}\]
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