Question

Please explain geometric and harmonic progressions?

Answer 1

Arithmetic progression: `a_n = A+nd`

Geometric progression: `a_n = An^q`

Harmonic progression: `a_n = 1/(A+nh)`

Explanation:

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:

`a_1, a_2,..., a_n,...`

such that the difference between two consecutive numbers is constant:

`a_n- a_(n-1) = d` for every `n`

If we define:

`A = a_1-d`

then we have:

`a_1 = A+d`

`a_2-a_1 = d => a_2 = A+2d`

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

`a_n = A+nd`

If we consider three consecutive terms we have:

`(a_(n-1) + a_(n+1))/2 = ((a_n-d) + (a_n+d))/2 = (2a_n)/2 = a_n`

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

Geometric progression

A geometric progression is a sequence of numbers:

`a_1, a_2,..., a_n,...`

such that the ratio between two consecutive numbers is constant:

`a_n/a_(n-1) = q` for every `n` and with `q!=0`

If we define:

`A = a_1/q`

then we have:

`a_1 = Aq`

`a_2/a_1 = q => a_2 = Aq^2`

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

`a_n = Aq^n`

If `q >0` and we consider three consecutive terms we have:

`sqrt(a_(n-1)a_(n+1)) = sqrt((a_n/q)(a_n*q)) = sqrt(a_n^2) = a_n`

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

Harmonic progression

A harmonic progression is a sequence of numbers:

`a_1, a_2,..., a_n,...`

such that their reciprocal constitute an arithmetic progression

`1/a_n-1/a_(n-1) = h` for every `n`

If we define:

`A = 1/a_1-h`

then we have:

`a_1 = 1/(A+h)`

`1/a_2-1/a_1 = h => 1/a_2 = A+2h`

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

`a_n = 1/(A+nh)`

If we consider three consecutive terms we have:

`2/(1/a_(n+1) +1/(a_(n-1))) = 2/((1/a_n+h)+(1/a_n-h))= 2/(2/a_n) = a_n`

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