Please explain geometric and harmonic progressions?

1 Answer
Feb 5, 2017

Arithmetic progression: a_n = A+ndan=A+nd

Geometric progression: a_n = An^qan=Anq

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