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Definitions : Sequence, Geometric Progression, String, Recursive

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On the following terms could you please give my an English text description - in your own words. Thanks.

1. Sequence
2. Geometric Progression
3. String
4. Recursive definition of a function
5. Recursive definition of a set
6. Recursive algorithm
7. Program correctness
8. Loop invariant
9. Final assertion

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Definitions of Sequence, Geometric Progression, String, Recursive definition of a function, Recursive definition of a set, Recursive algorithm, Program correctness, Loop invariant, Final assertion are provided. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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On the following terms could you please give my an English text description - in your own words. Thanks.

1. Sequence
2. Geometric Progression
3. String
4. Recursive definition of a function
5. Recursive definition of a set
6. Recursive algorithm
7. Program correctness
8. Loop invariant
9. Final assertion

Sequence: In mathematics, a sequence is a list of objects (or events) arranged in a "linear" fashion, such that the order of the members is well defined and significant.

In other words, a sequence can be defined by a function f: N  S, where N is set of natural numbers and S is any abstract set.

With f(1) = f1, f(2) = f2, ......... f(n) = fn
And the sequence is represented as: {f1,f2,f3,.........fn}

NOTE: If the letter n is finite, then the sequence is called as finite sequence, otherwise if n is infinite, and then the sequence is an infinite sequence.

Geometric progression: It is also called as geometric sequence, which is a sequence of numbers such that the quotient of any two successive members of sequence is a constant called common ratio of the sequence.

Thus, we have a geometric sequence written as

a1 = ar0, a2 = ar1, a3 = ar2,........., an = arn-1......, where r is the common ratio, a ...

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