Recursive Sequences
Read this topicExplanations 6
- Think about this sequence: 1,1,2,3,5,8,…1,1,2,3,5,8,\dots1,1,2,3,5,8,… This is the famous Fibonacci Sequence. This sequence is a perfect example of a recursive sequence. A recur…
By Hyunwoo (David) Lee, Expii Team, Moor Xu, Victor Wang, Ruiran Xun
- Write out the terms of the recursive sequence for an=an−12−3a_n=a_{n-1}^2-3an=an−12−3 for a1=1a_1=1a1=1 from 111 to 666. You might notice a pattern! If an=an−ka_n=a_{n-k}an=…
By Wayne Zhao, Ruiran Xun, Anusha Rahman
- In order to precisely define a recursive sequence, we need two parts: the relationship relating the next term to previous terms, and as many original terms as are needed to plug…
By Wayne Zhao, Ruiran Xun
- Recursive Sequences Our first lesson in sequences will be about recursive sequences! A recursive sequence is a type of sequence where the next term depends on the previous term.…
By Anusha Rahman
- (Video) Recursive Sequences by PatrickJMT In this video by PatrickJMT, you will walk through an example of recursive sequences. Summary This video explains how to find the first…
By Anusha Rahman
- Recursive Sequences A recursive sequence is a list of numbers where each number depends on one (or more) number that comes before it. Instead of giving a formula to jump at any …
By Anusha Rahman
Problems 16
- Let a1,a2,…a_1,a_2,\ldotsa1,a2,… be a sequence defined by a1=1a_1 = 1a1=1 and for n≥1n\ge1n≥1, an+1=an2−2an+3+1a_{n+1} = \sqrt{a_n^2 -2a_n + 3} + 1an+1=an2−2an+3+1. Find …
- What is B−AB - AB−A?
- If, for some positive real numbers a,b,c,da,b,c,da,b,c,d, the terms log2a,log2b,log2c,log2d\log_2 a, \log_2 b, \log_2 c, \log_2 dlog2a,log2b,log2c,log2d form an arithmet…
- We can define a sequence as follows: a0=−1a_0 = -1a0=−1, and for n≥1n \ge 1n≥1, an=(an−1)na_n = (a_{n-1})^nan=(an−1)n. What is a99a_{99}a99?
- 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…
- 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…
- Let x1,x2,…,x10x_1,x_2,\dots,x_{10}x1,x2,…,x10 be 101010 numbers. Suppose that xi+2xi+1=1x_i + 2 x_{i + 1} = 1xi+2xi+1=1 for each iii from 111 through 999. What is the valu…
- The Fibonacci numbers are defined recursively by the equation Fn=Fn−1+Fn−2F_n = F_{n - 1} + F_{n - 2}Fn=Fn−1+Fn−2 for every integer n≥2n \ge 2n≥2, with initial values F0=0F_0…
- Let f1f_1f1, f2f_2f2, f3f_3f3, ..., be a sequence of numbers such that fn=fn−1+fn−2f_n = f_{n - 1} + f_{n - 2}fn=fn−1+fn−2 for every integer n≥3n \ge 3n≥3. If f7=83f_7 = 8…
- Define a sequence such that a1=1a_1 = 1a1=1 and for all n≥2n \ge 2n≥2, an=(an−1)na_n = (a_{n-1})^nan=(an−1)n. What is a274a_{274}a274?
- Fix x∈Rx \in \mathbb{R}x∈R and define a series recursively as follows: {a0=(x−1)an=(an−1)(an−1+2)for n≥1\begin{cases} a_0 = (x-1) \\ a_n = (a_{n-1})(a_{n-1} + 2) & \text{for } n…
- Let octoπocto\pioctoπ be the first surface area “tau” (constant) value that is divisible by π8\pi^8π8. What is the exact value of this constant? For this problem, you will need …
- Let us define a sequence a0,a1,a2,⋯a_0,a_1,a_2,\cdotsa0,a1,a2,⋯ such that a0=0a_0 = 0a0=0, a1=1a_1 = 1a1=1, and an+2=2an+1+ana_{n+2} = 2a_{n+1}+a_nan+2=2an+1+an for n≥2n…
- What is the value of ∑n=010an\sum\limits_{n=0}^{10} a_nn=0∑10an?