For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

Sequences and series are often considered as the fourth part of calculus, in addition to limits, differentiation and integration.

Sequences are rows of real numbers. There are special types of sequences, such as arithmetic sequences (or arithmetic progressions), where the next term is a constant more than the previous. In geometric progressions the next term is a constant multiplied by the previous term.

A sequence can be given by a direct formula, or by a recurrence relation. In a recurrence relation the relation between the next term and the earlier terms are given. An example is the recurrence relation $F_{n+2}=F_{n+1}+F_n$. Together with the initial terms $F_1=F_2=1$ this recurrence relation forms the famous Fibonacci sequence.

A series is formed by summing a sequence. One of the questions is here: When the number of summed terms goes to infinity, does the sum approach a finite limit? In other words, is is it convergent? Several tests, such as the ratio test and the root test can help to answer these questions.

Important note. Questions about guessing the next number in a sequence, with no explicit mathematical context, will usually be quickly closed. Consider posting this question at puzzling.SE instead.

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