39takin m2<ak<m2+2m+1
we have s(ak)= ak -m2 not equal to zero.
Not the kind of convergence that appears in undergrad courses. So please dont be scared away. In fact if you apply your mind to this one, you may be rewarded.
For every integer n \ge 0 , let S(n) = n - m^2 where m is the greatest integer with m^2 \le n.
Define a sequence (a_k)_{k=0}^{\infty} by a0 = A, where A is a natural number and ak+1 = ak + S(ak).
For what values of A is the sequence eventually constant?
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42 Answers
62it is not true..
Let me give an example
suppose hypothetically that ak was 9900 at some stage and s(9900) becomes 100
then ak+1 becoems 10,000
which si perfect square..
how do u rule out such a possibility?
24Sorry If I am being rude[2][2] to the highly genious theprophet sir but couldn't resist....
If my memory is not that much weak then you only wrote this ::
""Mathematician, such questions should be posted in Olympiad Corner. Otherwise it will alarm the students that is in JEE syllabus."" in this link http://targetiit.com/iit_jee_forum/posts/inequalities_3084.html
Then have u forgotten ur advice or just by mistake posted this question here??????????????[7][7][7][7][7][7]
Again being sorry for writing all this
341@eureka: Since I am not a teenager I think you may safely assume that I will be responsible enough to post problems in the appropriate forums. Also since I have solved the problem myself, I know its difficulty level (and Nishant Sir who has also cracked it, will agree on this point). I dont have the habit of posting problems whose solution I have read up somewhere, and that is out of this same sense of responsibilty.
To the rest, who have an eye on doing well in the coming JEE, a little bit of an off-beat problem, which will stretch your mind a bit, is what you will need in the run-up to JEE. So give it a try.
1well either i din get d question.. or am a perfect dumb...
how can there be any A for which the whole sequence is constant?
for any A however, the sequence is constant upto third term...
24@theprophet:I never intended to say that you are an irresponsible user of this community.Through my post ,I only wanted to ask the reason because of which this question was posted here.
And Nishant sir has been kind enough to help me understand the toughness level of the problem and its priority level.....the same point being repeated by you in post#16.
I have read your solutions in the forum and from bottom of my heart appreciate them and your innovative techniques.
And in post#3 I have already tendered an apology from my side if you felt that whatever I wrote was meant to defame you or disregard you.
Hope this small spat wouldn't spoil our relations ...and wouldn't affect the environment of the forum which has been outstanding.
39SIR WAT ABOUT MY SOLN?
I GOT THAT ANSWER AS `ALL PEFECT SQUARES`
BUT NISHANT BHAIYAN IS ASKING ME TO CLAIM FOR NON SQUARES
341@star/sky - it need not be constant right from the beginning. Eventually constant, means after some particular term, all terms are equal.
Its obvious that for perfect square A, the sequence is constant. Thats why Nishant is saying prove that there are no non-perfect squares for which it does not become constant
39then in watever i am proving regarding perfect squares, where am i going wrong?