Coming up with recursive and closed form expressions

In summary, for the first sequence, the closed form expression is Cn = (1/4) * (1/3)^n-1 and the recursive formula is Cn = Cn-1/3. For the second sequence, the closed form expression is Dn = n! and the recursive formula is Dn = n*Dn-1. For the third sequence, the closed form expression is (3n-2)/(2n+1) and for the fourth sequence, the recursive formula for Qn is Qn = Qn-1 + Qn-2.
  • #1
KevinL
37
0

Homework Statement



I am having some trouble coming up with recursive and closed form expressions of different sequences. I realize helping me with this would pretty much just be giving me the answer, but if anyone could also help me with how to think of the answers that would be nice.

1) Cn = (1/4, 1/12, 1/36, 1/108)

CF: ?
R: C(n-1)/3

2) Dn = (1, 2, 6, 24, 120)

CF: ?
R: n*D(n-1)

3) I only need the closed form for this. (1/3, 4/5, 7/7, 10/9, 13/11)

CF: ?/2n+1

4) Let (1, 1, 2, 3, 5, 8) be the Fibonacci sequence. Define a new sequence by Qn = F(n+1)/Fn

a. List the first several terms of Qn
(1, 2, 3/2, 5/3, 8/5)

b. Find a recusive formula for Qn

?
 
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  • #2


Let's start with the first one. You correctly noticed that the sequence is given by a recursion relation
[itex]C_0 = 1/4, C_n = C_{n - 1} / 3[/itex]
(note how I wrote down explicitly what [itex]C_{n-1}/3[/itex] gives you by using an equality sign, and that I have included the first term which you need to calculate anything using the recursion relation).

Now make a few steps in your mind. If you want to calculate the second term in the series, you have to take the first one, 1/4, and divide it by 3. To go to the third, take the second, (1/4)/3 = 1/12, and divide it by 3. Now how can I go directly from the first term, 1/4, to the third one, 1/36?
Suppose that I want to go from the first one to the fifth...
 

FAQ: Coming up with recursive and closed form expressions

1. What is the difference between a recursive and a closed form expression?

A recursive expression is a mathematical formula that uses previous terms in the sequence to generate a new term. A closed form expression, on the other hand, is a formula that directly calculates the value of a term without using any previous terms.

2. When should I use a recursive expression versus a closed form expression?

Recursive expressions are useful when the formula for a term is dependent on previous terms and can lead to a more efficient computation. Closed form expressions are useful when the formula for a term can be directly calculated without relying on previous terms.

3. How do I come up with a recursive expression for a sequence?

To come up with a recursive expression, you need to identify the pattern in the sequence and determine how each term is related to the previous terms. Once you have the pattern, you can write a formula that uses the previous terms to generate a new term.

4. Can a recursive expression be converted into a closed form expression?

Yes, in most cases, a recursive expression can be converted into a closed form expression. This is done by solving the recursive formula for the nth term and simplifying it to a closed form expression.

5. Are recursive and closed form expressions only used in mathematics?

No, recursive and closed form expressions are used in many fields, including computer science, physics, and economics. They are particularly useful in situations where a sequence or pattern needs to be modeled or analyzed.

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