Recursive Formulas - Definition & Examples
Read this topicExplanations 4
- (Video) Recursive Sequences by PatrickJMT This video by PatrickJMT talks about recursive sequences. Summary\large{\text{Summary}}Summary A recursive sequence is kind of like a s…
By Stefan Cuevas, Alex Federspiel, Joshua Siktar, Amy Feraco, Jonathan Heller, Caroline K
- Recursive Formulas Recursive formulas seem to show up everywhere we look, and they can be surprisingly useful. However, they can also be frustratingly mysterious. A recursive fo…
By Mark Paulson, Alex Federspiel, Victor Wang, Joshua Siktar, Jonathan Heller, Caroline K
- Maximum Depth Exceeded: Fun with Sequences and Python, Part 1 We're going to talk about sequences, specifically about the relationship between closed forms and recursive formula…
By Joshua Siktar, Caroline K
- Recursive Sequences Recursive sequences are sequences defined by recursive formulas. They are also called recurrence relations or recurrence equations. A recursive formula relat…
By Caroline K, Jonathan Heller
Problems 30
- Suppose a1=2a_{1} = 2a1=2 and a2=3a_{2} = 3a2=3. We have a sequence that is described by an=an−1+2an−2a_n = a_{n-1} + 2a_{n-2}an=an−1+2an−2. What is a5a_{5}a5?
- Suppose a1=2a_{1} = 2a1=2 and a2=4a_{2} = 4a2=4. If the sequence is defined by the recursive formula an=an−1⋅an−2a_{n} = a_{n-1} \cdot a_{n-2}an=an−1⋅an−2, what is a4a_{4}a4?
- Given the recursive formula an=2an−1a_{n} = 2^{a_{n-1}}an=2an−1 and a1=2a_{1} = 2a1=2, what is a4a_4a4?
- Given the recursive formula an=2an−1a_{n} = 2^{a_{n-1}}an=2an−1 and a1=2a_{1} = 2a1=2, what is a3a_3a3?
- If a sequence a1,a2,⋯a_1,a_2,\cdotsa1,a2,⋯ is defined by the recursive formula an=n⋅an−1a_{n} = n \cdot a_{n-1}an=n⋅an−1 and a1=1a_{1} = 1a1=1, find a2a_2a2, a3a_3a3, and…
- A sequence sss is described by the formula sn=f(n)g(n)s_n = f(n) g(n)sn=f(n)g(n), where f(n)f(n)f(n) and g(n)g(n)g(n) are recursively defined as follows: f(n)=(n−1)f(n−1),f(1)=…
- Convert the following recursive formula for h(n)h(n)h(n) to an explicit formula. h(0)=1h(0) = 1h(0)=1 h(1)=1h(1) = 1h(1)=1 h(n)=n⋅h(n−1)h(n) = n\cdot h(n-1)h(n)=n⋅h(n−1)
- Convert the following recursive formula for g(n)g(n)g(n) to an explicit formula. g(0)=0g(0) = 0g(0)=0 g(1)=1g(1) = 1g(1)=1 g(n)=n+g(n−1)g(n) = n + g(n-1)g(n)=n+g(n−1)
- Convert the following recursive formula for g(n)g(n)g(n) to an explicit formula. g(0)=0g(0) = 0g(0)=0 g(1)=1g(1) = 1g(1)=1 g(n)=n2+g(n−1)g(n) = n^2 + g(n-1)g(n)=n2+g(n−1)
- Find the recursive formula for 0.3,0.03,0.003,0.0003,⋯0.3,0.03,0.003,0.0003, \cdots0.3,0.03,0.003,0.0003,⋯
- Samantha has 2000\20002000 to spend on food, which she saved up while she lived with her parents. Samantha adds 100\100100 to her monthly food savings each month. How many month…
- We can define a sequence as follows: a0=0a_0 = 0a0=0, and for n≥1n \ge 1n≥1, an=an−1+2a_n = a_{n-1} + 2an=an−1+2. What is a2016a_{2016}a2016?
- Suppose that we have a sequence which satisfies, for n≥1n \ge 1n≥1, an=3an−1−4a_n = 3a_{n-1} - 4an=3an−1−4. If a4=83a_4 = 83a4=83, what is a0a_0a0?
- Suppose we have a sequence which, for n≥1n \ge 1n≥1, satisfies an=n⋅(an−1)a_n = n \cdot (a_{n-1})an=n⋅(an−1). If a0a_0a0 is an integer, which of the following is not a possib…
- Let's try a sequence similar to the Fibonacci numbers! If a0=0a_0 = 0a0=0, a1=1a_1 = 1a1=1, and for n≥2n \ge 2n≥2, an=2(an−1)+an−2a_n = 2(a_{n-1}) + a_{n-2}an=2(an−1)+an−2,…
- Suppose that we have a sequence where an=11+an−1a_n = \frac{1}{1+a_{n-1}}an=1+an−11. If a3=2a_3 = 2a3=2, what is the value of a−1a_{-1}a−1, as a decimal?
- If Abrim gave a third of his pencils away every day, what recursive rule describes ana_nan, the number of remaining pencils on day nnn?
- What is a5a_5a5?
- What is a5a_5a5?
- Consider the recursive sequence starting at a1a_1a1 and defined by a1=1a_1 = 1a1=1 and an=k⋅an−1a_n = k\cdot a_{n-1}an=k⋅an−1, where kkk is a natural number. Which of the fo…
- A sequence a1,a2,⋯a_1,a_2,\cdotsa1,a2,⋯ is defined by the recursive formula an=2+an−1an−1a_n=2+a_{n-1}^{a_{n-1}}an=2+an−1an−1 and a1=1a_1=1a1=1. Find a3a_3a3.
- What is the value of a4a_4a4?
- What is the value of a4a_4a4?
- Write the sequence 1,4,16,64,⋯1,4,16,64,\cdots1,4,16,64,⋯ as a recursive sequence where n≥1n\ge1n≥1.
- A sequence is given by 0,1,3,6,10,⋯0,1,3,6,10,\cdots0,1,3,6,10,⋯ where n≥1n\ge1n≥1. Write this as a recursive sequence.
- Define a sequence {an}\{a_n\}{an} such that a1=a2=1,a3=2a_1=a_2 = 1, a_3 = 2a1=a2=1,a3=2 and an+an−3=an−1+an−2a_n + a_{n-3} = a_{n-1} + a_{n-2}an+an−3=an−1+an−2 for n>3n…
- Define a sequence {an}\{a_n\}{an} by a1=a2=1,a3=2a_1=a_2=1, a_3 = 2a1=a2=1,a3=2 and an+an−3=an−2+an−1a_n + a_{n-3} = a_{n-2}+a_{n-1}an+an−3=an−2+an−1 for n>3n > 3n>3. Fi…
- Let f1f_1f1 denote the string "EXPII". The string fk+1f_{k+1}fk+1 is created by inserting the string EXPII\text{EXPII}EXPII in between each of it's letters. For example, f2=EE…
- The factorial "!" is defined for integers nnn such that n!=n(n−1)(n−2)⋯1n! = n(n-1)(n-2) \cdots 1n!=n(n−1)(n−2)⋯1. For example, 5!=5⋅4⋅3⋅2⋅1=1205! = 5 \cdot 4 \cdot 3 \cdot 2 \c…
- For n≥2n \ge 2n≥2 which of the following is equal to Fn+12−Fn−12F_{n+1}^2 - F_{n-1}^2Fn+12−Fn−12?