IBM Research | Ponder This | May 2003 challenges

# Ponder This

## May 2003

Ponder This Challenge:

This month's puzzle was suggested by Venkateshwar Rao Thota, based on a result of Srinivasa Ramanujan.

I will use "sqrt" to denote the positive square root, "b*c" to denote multiplication, and "b**c" to denote exponentiation (b raised to the c power).

Define f(n)=sqrt(1+2*sqrt(1+3*sqrt(1+4*sqrt(1+5*sqrt(...+n*sqrt(1)...))))) So f(1)=sqrt(1)=1, f(2)=sqrt(1+2)=sqrt(3)=1.732..., f(3)=sqrt(1+2*sqrt(1+3))=sqrt(1+4)=sqrt(5)=2.236... .

Clearly f is an increasing function of n.

First part: Give (with proof) a bound B such that f(n)<B for all n.

Second part: Give (with proof) an integer n such that f(n) > B - 1/10**100, that is, f(n) agrees with B up to 1 in the 100th decimal place.

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Challenge: 05/01/2003 @ 11:00 AM EST
Solution: 06/02/2003 @ 4:00 PM EST
List Updated: 06/23/2003 @ 5:00 PM EST

Joe Fendel (5.01.2003 @03:52:13 PM EDT)

Rocke Verser (5.01.2003 @04:15:06 PM EDT)
Oleg Mazurov (5.01.2003 @05:47:02 PM EDT)
Vitaly Stakhovsky (5.01.2003 @05:47:02 PM EDT)
Alexey Vorobyov (5.01.2003 @09:02:15 PM EDT)
Ling Li (5.01.2003 @09:59:53 PM EDT)
Hagen von Eitzen (5.02.2003 @12:04:34 PM EDT)
Yunhong Zhou (5.03.2003 @04:09:23 AM EDT)
Surendar Reddy Nanchary (5.02.2003 @03:47:46 PM EDT)
Michael Brand (5.04.2003 @06:39:00 AM EDT)
Klaus Hoffmann (5.03.2003 @03:47:58 PM EDT)
Gyozo Nagy (5.04.2003 @03:36:10 AM EDT)
JoséH. Nieto (5.04.2003 @09:50:01 AM EDT)
Venezuela Maracaibo (5.04.2003 @09:50:01 AM EDT)
Ignacio Larrosa Cañestro (5.04.2003 @06:14:34 PM EDT)
Coserea Gheorghe (5.05.2003 @02:56:54 AM EDT)
George J. Ross (5.05.2003 @10:44:10 AM EDT)
Paul Lee (5.05.2003 @07:28:35 PM EDT)
David Woodruff (5.06.2003 @02:26:43 AM EDT)
Nis Jorgensen (5.06.2003 @04:44:39 AM EDT)
Abhinav Kumar (5.07.2003 @01:51:39 AM EDT)
John G. Fletcher (5.07.2003 @08:42:06 PM EDT)
Ariel Flat (5.08.2003 @01:51:18 PM EDT)
Alex Proudfoot (5.09.2003 @03:26:29 AM EDT)
Shmuel Spiegel (5.09.2003 @04:19:06 AM EDT)
Karl Dsouza (5.11.2003 @01:01:18 AM EDT)
Amit Sinha (5.12.2003 @03:27:42 AM EDT)
Andreas Knappe (5.12.2003 @04:45:35 AM EDT)
Jitesh Jain (5.15.2003 @03:44:01 AM EDT)
R. Nandakumar (5.20.2003 @08:40:23 AM EDT)
Raghav Boinepalli (5.21.2003 @02:13:57 PM EDT)
Daniel Chong Jyh Tar (5.21.2003 @09:41:56 PM EDT)
Maxim G. Smith (5.22.2003 @03:19:49 PM EDT)
Dave Blackston (5.23.2003 @01:11:37 PM EDT)
Wei Liu (5.23.2003 @03:09:13 AM EDT)
Florbela Duarte (5.26.2003 @07:20:32 PM EDT)
Vishal Goel (5.30.2003 @01:20:25 AM EDT)
Alexis G Megas (5.30.2003 @06:22:57 PM EDT)
Dan Dima (6.1.2003 @05:50:12 AM EDT)
Dan Yu (6.2.2003 @04:42:50 AM EDT)
Andre Rzym (6.2.2003 @03:02:13 PM EDT)

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