A Brief History about the Harmonic Sequence

Harmonic Series was first proven in the 14th century by Nicole Oresme, but this achievement fell into obscurity. Proofs were given in the 17 th century by Pietro Mengoli, Johann Bernoulli, and Jacob Bernoulli.

Harmonic sequences have had a certain popularity with architects, particularly in the Baroque period, when architects used them to establish the proportions of floor plans, of elevations, and to establish harmonic relationship between both interior and exterior architectural details of churches and palaces.

A violin’s first harmonic tone is obtained by lightly plucking the string at its midpoint. The second harmonic tone is obtained by plucking it one-third the way down and so on. In mathematical form this example is shown as:

A n example of a harmonic sequence is…

Notice that the reciprocals of the terms form the arithmetic sequence…

2, 3, 4, 5, …, n, …

What is Harmonic Mean?

Harmonic means are terms that are between any two nonconsecutive terms of a harmonic sequences.

Below is an example of a harmonic mean…

Why?Because 1/3 is between ½, ¼.

Below is an example of a harmonic mean…

Why?Because 1/3 and ¼ is between ½, 1/5.

How can we determine the nth term of a harmonic sequence?

Consider the reciprocals of the given terms, then find the nth term of the resulting arithmetic sequence, and then take its reciprocal.

Let’s practice, shall we?

Find the 10th term of the harmonic sequence

10th term of

Get the reciprocal: 2, 4, 6, 8

Use the formula an = a1 + (n – 1)d

10th term of

Substituting we have… an

= 2 + (10 – 1)2 = 2 + (9)2 = 20

Therefore, is the 10th term of the harmonic sequence

Insert three harmonic means between…

three harmonic means between…

Reciprocal of the following: 4 and 20

Substituting these values in the arithmetic formula we have 20

= 4 + (5 – 1)d = 4.

three harmonic means between…

We now have the common difference d which is 4. Now we simply add 4 to the other values to get the harmonic means.

three harmonic means between…

Therefore, the harmonic means are

Think about this…Interesting number patterns are all around us. For example, the scales of a pineapple form a double set of spirals – one going clockwise, and one going counterclockwise. When we count theses spirals, we see three distinct families of spirals with usually 5, and 8, or 8 and 13, or 13 and 21 spirals.Can you see a specific pattern in the sequence of numbers 5, 8, 13, 21,…?

The numbers 5, 8, 13, 21,… are called Fibonacci numbers. They are terms of the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…

A Brief History about the Fibonacci sequence

Fibonacci Series is a series of numbers in which each member is the sum of the two preceding numbers. For example, a series beginning 0, 1 ... continues as 1, 2, 3, 5, 8, 13, 21, and so forth. The series was discovered by the Italian mathematician Leonardo Fibonacci (circa 1170-c. 1240), also called Leonardo of Pisa. Fibonacci numbers have many interesting properties and are widely used in mathematics. Natural patterns, such as the spiral growth of leaves on some trees, often exhibit the Fibonacci series.

Leonardo FibonacciFounder of the Fibonacci

sequence

Find the sum of the first five odd terms of the Fibonacci sequence; that is, F1 + F3+ F5 + F7 + F9.

F1 + F3+ F5 + F7 + F9 (Remember the basic pattern)

= 1 + 2 + 5 + 13 + 34

= 55

The sum of the first n odd terms of the Fibonacci sequence is F2n; that is, F1 + F3 + … + F 2n-1 = F2n.

Remember that…A sequence of numbers whose reciprocals form an arithmetic

sequence is called a harmonic sequence.The terms between any two nonconsecutive terms of a

harmonic sequence are called harmonic means.A sequence of numbers in which the first two terms are 1 and

each terms is the sum of the preceding terms is called Fibonacci sequence.

If you are patient in one moment of anger, you will escape a hundred days of sorrow.

Chinese Proverb