Please explain geometric and harmonic progressions?

Question

Christopher Allers · Accepted Answer

Arithmetic progression: #a_n = A+nd#

Geometric progression: #a_n = An^q#

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

Isabella Ackerman · Answer

undefined 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.