341or if you consider f(n) = \frac{1}{2 + \frac{1}{n}}, its easy to see that it increases with n.
Without calculating, which is greater?
\frac{361254001}{722508003} or \frac{361254002}{722508005}
If I remember correctly, this is one of the first things i learnt from Hall & Knight
Edit:
Hint: if A/B< C/D, what can you say about A+CB+D
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26 Answers
1but how can we use the above fact as
725008003≠2n+1 where n=361254001
3411got it
a/b<c/d
ad<bc
adding ab to both sides
(a+c)/(b+d)>a/b
a=361254001 , b=725008003 , c=1 , d=2
the result follows
62you are so close fibonacci..
you have to identify a, b, c, and d :P
1from nishant sir's hint
a/b<c/d
ad<bc
adding cd to both sides
(a+c)/(b+d)<c/d
how can we prove the inequality from this result
1set 361254001=n and 725008004=x
assume n/(x-1) > (n+1) / (x+1)
or nx+n > nx+x-n-1
or 2n>x-1
which is a contradiction
so 361254001/725008003 < 361254002/725008005
62try a bit more :)
and see why this is relevant here ;)
I am sure you will be able to reason out this question yourself...
11Sir,plese solve the problem according to your hint.
abhi bhi hence proved nehi hua.
1Hence Proved ........ ;-)
Sir please keep posting such questions....these are really entertaining.
62The answer is correct and so is teh reasoning..
seee my hint for a general view on the same question :)
11361254002/725008005>361254001/725008003
explanation:If we prove that;
361254002*725008003>361254001*725008005
then we are done.
now if both 725008003 and 725008005 were multiplied by 361254001 then 725008005* ".." would be greater than 725008003* ".." by 722508002(361254001*2).
Now,this is less than 725008003.
Thus,725008003* "..">725008005* ".."
1well eureka has hidden this answer.....but i can understand in which way he is trying to do it....i'll definitely try this out...
62@swastika... the problem with calculator is that the accuracy is lower than what we want for this problem ;)
@eureka.. yes it is correct
@Aveek.. I dont see any logic
24@sir
I would love to see the simpler logic.[1][1]
but this method is fine naa?
3Sir, is the second one greater !! It is 361254002 parts out of 725008005 .
But with calculator both the values are coming to be 0.49827588.
COULDN'T Understand!!
62@aveek: do you think your first assertion is correct?
@Eureka..: there is a far simpler logic...
24easy...
361254002725008005=361254001+1725008003+2
361254001725008003-361254001+1725008003+2
=361254001.(725008003+2)-(361254001+1)725008003(725008003)(725008003+2)
=361254001.2-361254001.725008003(725008003)(725008003+2)
=361254001(-......)(725008003)(725008003+2)<0
=>361254001725008003-361254001+1725008003+2<0
=>361254001725008003-361254002725008005<0
=>361254001725008003 <361254002725008005
I dont know whether I am allowed to post the answer..so I am hiding it[1]
1I think 361254002725008005 will be greater as
361254001725008003 <361254002725008004
and
361254002725008004 >361254002725008005 .........right ??? :-(
62no they are not... think again! or else use the calculator to check ;)