All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Numbers
An interesting sequence (Posted on 2012-11-10) Difficulty: 3 of 5
(a) What is interesting about the following sequence? 1/89, 1/9899, 1/998999, 1/99989999, ...
(b) Explain.

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 4.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution The formula Comment 2 of 2 |

x=1/(10^k(10^k-1)-1), for all terms.

The first term, 1/89, is sum (1 to infinity) F(n)/10^(n+1), where F represents the Fibonacci numbers, see http://mathworld.wolfram.com/FibonacciNumber.html

If we then look at the second term:1/9899 =0.0001010203050813213455904636832003232649762602283058; and the third: 1/998999 = 1.0010020030050080130210340550891442333776109885*10^-6 etc. with the Fibonacci numbers stretched out a bit further each time.

Using Livio's relation (19) from the cited text, the expression:
Sum 1 to infinity x*fibonacci(n)*x^n
is reached, where x = 1/10, 1/100, 1/1000 etc.

Generalising, sum n=1 to infinity 1/(10^k)*fibonacci(n)*1/(10^k)^(n) = 1/(10^k(10^k-1)-1), for all terms, since:

1/(10^k(10^k-1)-1) = 1/(10^(2k)-10^k-1), and also
sum_(n=1)^infinity F_n/(10^k (10^k)^n) = 1/(10^(2k)-10^k-1)

Additional note:

After checking, the significant result is that 1/89 is sum (1 to infinity) F(n)/10^(n+1). Given this (or any other infinite sum which happens to produce a rational 2-digit fraction of the form 1/xx) the rest follows automatically as it amounts merely to an instruction to flank the appropriate fraction with increasing numbers of 9s. So perhaps not that interesting after all...

Edited on November 11, 2012, 7:58 am
  Posted by broll on 2012-11-11 01:58:20

Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (4)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information