We consider a sequence of words consisting of the letters “A” and “B”. The first word in the sequence is “A”, the \(k\)-th word is obtained from the \((k-1)\)-th by the following operation: each “A” is replaced by “AAB” and each “B” by “A”. It is easy to see that each word is the beginning of the next, thus obtaining an infinite sequence of letters: AABAABAAABAABAAAB...
a) Where in this sequence will the 1000th letter “A” be?
b) Prove that this sequence is non-periodic.
There are one hundred natural numbers, they are all different, and sum up to 5050. Can you find those numbers? Are they unique, or is there another bunch of such numbers?
What’s the sum of the Fibonacci numbers \(F_0+F_1+F_2+...+F_n\)?
What’s the sum \(\frac{F_2}{F_1}+\frac{F_4}{F_2}+\frac{F_6}{F_3}+...+\frac{F_{18}}{F_9}+\frac{F_{20}}{F_{10}}\)?
We have a sequence where the first term (\(x_1\)) is equal to \(2\), and each term is \(1\) minus the reciprocal of the previous term (which we can write as \(x_{n+1}=1-\frac{1}{x_n}\)).
What’s \(x_{57}\)?
Let \(n\) be a positive integer. Can \(n^7-77\) ever be a Fibonacci number?
Prove that every pair of consecutive Fibonacci numbers are coprime. That is, they share no common factors other than 1.
Calculate the following: \(F_1^2-F_0F_2\), \(F_2^2-F_1F_3\), \(F_3^2-F_2F_4\), \(F_4^2-F_3F_5\) and \(F_5^2-F_4F_6\). What do you notice?