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Fibonacci Fractions (Posted on 2005-03-09) Difficulty: 3 of 5
What is the sum of 0.1+ 0.01+ 0.002+ 0.0003+ 0.00005+ 0.000008+ ..., where each term is the n-th Fibonacci number, shifted n places to the right (that is, divided by 10^n)?

See The Solution Submitted by e.g.    
Rating: 3.0000 (2 votes)

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Solution Puzzle Solution: Method I | Comment 15 of 17 |
(In reply to answer by K Sengupta)

Let S = 0.1+ 0.01+ 0.002+ 0.0003+ 0.00005+ 0.000008+ 0.000013+.....

Then,

10S = 1+ 0.1+ 0.02+ 0.003+ 0.0005+ 0.00008+ 0.00013+ .....

or, 10S - S- 1 =  0.01 + 0.001 + 0.0002 + 0.00003 + 0.000005
+ 0.0000008+ ...,

or, 9S -1 = S/10

or, (89/10)*S = 1

or, S = 10/89

Edited on May 1, 2008, 6:30 am
  Posted by K Sengupta on 2008-05-01 06:30:01

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