(a) What is interesting about the following sequence? 1/89, 1/9899, 1/998999, 1/99989999, ...
(b) Explain.

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