Can all recursive functions be written explicitly?

In general, a recursive function where f(n)=g(n,f(n−1),f(n−2),…) can not always be converted to an explicit form. On the other hand, a recursive function in a computer program can be converted to a non-recursive (iterative) function.

How do you define a recursive sequence?

A recursive sequence is a sequence in which terms are defined using one or more previous terms which are given. If you know the nth term of an arithmetic sequence and you know the common difference , d , you can find the (n+1)th term using the recursive formula an+1=an+d .

How do you define a sequence explicitly?

The term an is called the nth term of the sequence, or the general term of the sequence. An explicit formula defines the nth term of a sequence using the position of the term. A sequence that continues indefinitely is an infinite sequence.

Are all sequences recursive?

Certain sequences (not all) can be defined (expressed) in a “recursive” form. In a recursive formula, each term is defined as a function of its preceding term(s). [Each term is found by doing something to the term(s) immediately in front of that term.]

Are recursive sequences linear?

In mathematics and theoretical computer science, a constant-recursive sequence (also known as a linear recurrence sequence, linear-recursive sequence, linear-recurrent sequence, or C-finite sequence) is an infinite sequence of numbers in which each number in the sequence is equal to a linear combination of one or more …

Which function defines the sequence?

We have a function f (x) defined on some interval (K,∞) and the same formula is used to define a sequence: an = f (n) for integers n > K. This means that we are picking points from the graph of f to form the sequence.

How do you define a sequence?

A sequence is an ordered list of numbers . The three dots mean to continue forward in the pattern established. Each number in the sequence is called a term. In the sequence 1, 3, 5, 7, 9, …, 1 is the first term, 3 is the second term, 5 is the third term, and so on.

Is every recursively defined sequence convergent?

Theorem 1.1 (Monotonic Sequence Theorem). Every bounded, monotonic sequence converges. If we are given a sequence in a recursive formula, an D f . an1/, we will need to check that it is bounded, check that it is monotonic (increasing or decreasing), and then find any fixed points.

How is arithmetic sequence defined?

An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2.