How it Works › Forums › Exploration Problems (Interesting Math Problems!) › Exploring Recursive Sequences!
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Hey Math Circle Students!
These math problems are for 20 points and really develop your skills as mathematicians to understand higher order thinking beyond just math competitions and AMC 8/Mathcounts, etc. These focus on observing and proving patterns and making conjectures!
If you don’t understand one of the questions, feel free to post in the questions forum here
1. Consider the terms of the Fibonacci sequence in mod 2, mod 3, mod 5. Note any patterns that you see and if you can, prove that there must be a pattern and calculate the period of the pattern. (Hint: $F_{n}=F_{n1}+F_{n2}$ where $F_{n}$ represents the $n$th term of the fibonnaci sequence which is 1,1,2,3,5,8… where you add the two previous terms to get the next term).
NOTE: The period of the “pattern” is defined as the smallest $k$ such that $F_{n+k}=F_{n}$ in the modulus you are working in. If that is unclear, then consider the following pattern: $1,2,3,1,2,3,1,2,3…..$.
The period of this pattern is $3$ because the length of each cycle of the pattern is $3$.2. Consider the recursion $A_n=A_{n1}+A_{n2}+…A_1$ where $A_1=1$ and the $n$th term is the sum of all previous terms. Find what $A_n$ is and prove your result.
3. The Fibonacci recursion is given by $F_n=F_{n1}+F_{n2}$ and we have a closed form for $F_n$ in terms of $n$ is given by $F_n=\frac{(\frac{1+\sqrt{5}}{2})^n+(\frac{1\sqrt{5}}{2})^n}{\sqrt{5}}$. Use problem #7 of the handout and evaluate the roots to the quadratic: $n^2=n+1$. What do you notice?
If you discover a pattern or you manage to prove one of the problems, post your discoveries below! These are really openended so you can post any progress you have made on the problems!

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