Periodic Sequences, in Depth
Read this topicProblems 8
- Suppose we have a sequence and a function fff such that, for n≥1n \ge 1n≥1, an=f(an−1)a_n = f(a_{n-1})an=f(an−1). If a0=aka_0 = a_ka0=ak for some kkk, is this sequence neces…
- Suppose we have a sequence which, for n≥1n \ge 1n≥1, satisfies an=(an−1)2+(an−1)−4a_n = (a_{n-1})^2 + (a_{n-1}) - 4an=(an−1)2+(an−1)−4. If this sequence is periodic with fund…
- What is the fundamental period of the sequence defined by an=(−3)n3na_n = \frac{(-3)^n}{3^n}an=3n(−3)n?
- Consider the sequence given by an=rna_n = r^nan=rn for some real number rrr. Is there a value of rrr such that this sequence is periodic with fundamental period greater than 111?
- Define a sequence as the following: an=1a_n = 1an=1 if nnn is divisible by 333, otherwise an=0a_n = 0an=0. Is this sequence periodic?
- Suppose we have a function fff which satisfies f(8)=20f(8) = 20f(8)=20, f(20)=5f(20) = 5f(20)=5, f(5)=128f(5) = 128f(5)=128, and f(128)=1f(128) = 1f(128)=1. Now suppose we have …
- Define a sequence recursively by a1=3a_1 = 3a1=3, a2=5a_2 = 5a2=5, and an=an−1an−2a_n = \frac{a_{n-1}}{a_{n-2}}an=an−2an−1 for positive integers n≥3n \ge 3n≥3. What is a20…
- The sequence a0a_0a0, a1a_1a1, a2a_2a2, …\ldots… satisfies the recurrence equation an=2an−1−2an−2+an−3a_n = 2 a_{n-1} - 2 a_{n - 2} + a_{n - 3}an=2an−1−2an−2+an−3 for eve…