# Please explain geometric and harmonic progressions?

Arithmetic progression:

Geometric progression:

Harmonic progression:

We also have to introduce the arithmetic progression, since the definition of the harmonic progression requires it.

Arithmetic progression

An arithmetic progression is a sequence of numbers:

such that the difference between two consecutive numbers is constant:

If we define:

then we have:

and so on, so we can see that the terms of an arithmetic progression can be expressed in the form:

If we consider three consecutive terms we have:

so each term is the arithmetic mean of the terms adjacent to it.

Geometric progression

A geometric progression is a sequence of numbers:

such that the ratio between two consecutive numbers is constant:

If we define:

then we have:

and so on, so we can see that the terms of an arithmetic progression can be expressed in the form:

so each term is the geometric mean of the terms adjacent to it.

Harmonic progression

A harmonic progression is a sequence of numbers:

such that their reciprocal constitute an arithmetic progression

If we define:

then we have:

and so on, so we can see that the terms of an arithmetic progression can be expressed in the form:

If we consider three consecutive terms we have:

so each term is the harmonic mean of the terms adjacent to it.

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Geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Harmonic progression is a sequence of numbers where each term is the reciprocal of an arithmetic progression. In other words, the reciprocal of each term forms an arithmetic progression.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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