Problem 3.5 of A Concise Introduction to Pure Mathematics

Problem
The Fibonacci sequence starts with term 1, 1 and then proceeds by letting the next term be the sum of previous two terms. So the sequence starts 1, 1, 2, 3, 5, 8, 13, 21, 34, ... With this in mind, consider the decimal expansion of x = 100/9899: it is 0.010102030508132134... Note how the Fibonacci sequence lives inside this expansion. Can you explain this? Do you think it continues forever?
(Hint: First show that 100 + x + 100x = 10000x)


Solve
x = 100/9899 => 9899x = 100 => 10000x - 101x = 100 => 100 + x + 100x = 10000x

It is obvious that x < style="font-weight: bold;">.a1a2a3a4....., while all a's are between 0 and 9, then we can write above equation as
100 + (3) 0.a1a2a3.... + (2) a1a2.a3a4.... = (1) a1a2a3a4.a5a6....
100 = (1) a1a2a3a4.a5a6.... - (2) a1a2.a3a4.... - (3) 0.a1a2a3....
To make the right hand side sum up to 100:
a1 can only be 0, otherwise (1) will be bigger than 100
a2 can only be 1
a3 can only be 0 (a3 of (1) - a1 of (2) = 0, and a1 is already 0)
a4 can only be 1 (a4 of (1) - a2 of (2) = 0, and a2 is 1)
for the rest, we have a(n + 4) - a(n + 2) - a(n) = 0, i.e. a(n+4) = a(n + 2) + a(n)
if we consider every two decimal digit as a number, then the sequence follows Fibonacci sequence, but when the number is greater than 100, which occupies more than 2 digits, the sequence will no longer show up.