How do find the n-th term in a sequence?

misyjny76

misyjny76

Answered question

2022-09-23

How do find the n-th term in a sequence?

Answer & Explanation

nutnhonyl8

nutnhonyl8

Beginner2022-09-24Added 8 answers

It depends on the type of sequence.
If the sequence is an arithmetic progression with first term a 1 , then the terms will be of the form:
a n = a 1 + ( n - 1 ) b
for some constant b.
If the sequence is a geometric progression with first term a 1 , then the terms will be of the form:
a n = a 1 r n - 1
for some constant r.
There are also sequences where the next number is defined iteratively in terms of the previous 2 or more terms. An example of this would be the Fibonacci sequence:
0,1,1,2,3,5,8,13,21,34,55,89,...
Each term is the sum of the two previous terms.
The ratio of successive pairs of terms tends towards the golden ratio
ϕ = 1 2 + 5 2 1.618034
The terms of the Fibonacci sequence are expressible by the formula:
F n = ϕ n - ( - ϕ ) - n 5 (starting with F 0 = 0 , F 1 = 1 )
In general an infinite sequence is any mapping from S for any set S. It can be defined in any way you like.
Finite sequences are the same, except that they are mappings from a finite subset of N consisting of those numbers less than some fixed limit, e.g. { n : n 10 }

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