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[The rules of eNotes site require that one question must be based on one topic. I'll answer about arithmetic and geometric sequences.]

By the definition, an arithmetic sequence starts from any number `a_1 = a,` and each next term is obtained from the previous by adding some fixed number...

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Hello!

[The rules of eNotes site require that one question must be based on one topic. I'll answer about arithmetic and geometric sequences.]

By the definition, an arithmetic sequence starts from any number `a_1 = a,` and each next term is obtained from the previous by adding some fixed number `d` called a common difference. Thus `a_n = a + (n-1)d.` Geometric sequences have a similar definition, but the next term is obtained from the previous by *multiplication* by a fixed number and `a_n = a * d^(n-1).` Therefore the number `d` is called the common ratio.

Both arithmetic and geometric sequences may increase or decrease, but geometric sequence is not monotone if its common ratio is negative. Any increasing geometric sequence grows faster than any arithmetic sequence for large indices. Both have a closed form formula of the sum of several consecutive terms.