A Harmonic Sequence, in Mathematics, is a sequence of numbers a1, a2, a3,… such that their reciprocals 1/a1, 1/a2, 1/a3,… form an arithmetic sequence (numbers separated by a common difference). It is a progression formed by taking the reciprocals of an arithmetic progression. The sum of a sequence is known as a series, and the harmonic series is an example of an infinite series that does not converge to any limit. That is, the partial sums obtained by adding the successive terms grow without limit, or, put another way, the sum tends to infinity. Equivalently, a sequence is a Harmonic Sequence when each term is the harmonic mean of the neighboring terms. It is not possible for a harmonic sequence (other than the trivial case where a = 1 and k = 0) to sum to an integer. The reason is that, necessarily, at least one denominator of the progression will be divisible by a prime number that does not divide any other denominator. The harmonic mean of two terms of the harmonic sequence is the term halfway between the two original terms.
Given: 1, ½, 1/3, ¼…
Observe the denominators of each term in the sequence. Try to get the reciprocals. Do these form a sequence? If so, what type of sequence?
Example1:
1, ½, 1/3, ¼… The reciprocal form an arithmetic sequence In the harmonic sequence 2/3, ½, 2/5, 1/3, 2/7… We can say that ½ is the harmonic means between 2/3 and 2/5; ½, 2/5 and 1/3 are the harmonic between 2/3 and 2/7. 2/3+1/2+2/5+… is a harmonic series.
Example 2: Find the 12th term of the harmonic sequence 1/9, 1/12, 1/15... Note that the reciprocal forms an arithmetic sequence so we may first find the 12th term of the harmonic sequence which is 1/42
Example 3:
Insert two harmonic means between 6 and 3/2. We first find the arithmetic means between 1/6 and 2/3. Since we have to insert two terms, then we have n= 4, a1= 1/6 and a4= 2/3 Using the formula an= a1+(n-1)d, we can find the common difference. a4= a1+(n-1)d 2/3= 1/6+(4-1)d d=1/6
So the arithmetic means are:
a2= a1+d
a3=a2+d
=1/6+1/6
=1/3+1/6
=1/3
=1/2
The reciprocals of these two terms result in the harmonic means between 6 and 3/2 which are 3 and 2
Example 4: 1/3, 1/6, 1/9 , ... are examples of Harmonic Sequence.
If you take the reciprocal of each term from the above Harmonic Sequence, the sequence will become 3, 6, 9, which is an Arithmetic Sequence with a common difference of 3.
Example 5: Another example of Harmonic Sequence is 6, 3, 2. The reciprocals of each term are 1/6, 1/3, 1/2 which is an Arithmetic Sequence with a common difference of 1/6.
To find the term of Harmonic Sequence, convert the sequence into Arithmetic Sequence then do the calculations using the Arithmetic Sequence formulas. Then take the reciprocal of the answer in Arithmetic Sequence to get the correct term in Harmonic Sequence.
Exercises: 1.What will be the sum of first 100 terms of this series: 8,13,18,23,28,… Answer:
n/2 (2a+(n-1)d N=100, a=8, d=5 100/2 (2*8+ (100-1)5 = 50(16+495) = 25, 550
2. Find the sum of the harmonic series 3/2+6/7+3/5+...6/19. What did you find out when you explored the sum, we need to find the number of terms in this sequence. We can compute it by using the formula an=a1+(n-1)d.
We have a1=2/3, an= 19/6 and d=7/6-2/3 or ½ an=a1+(n-1)d so, 19/6=2/3=(n-1)1/2 1/2n=19/6-2/3+1/2 There are two missing terms, a4 and a5. a4=a3+d
a5=a4+d
=5/3+1/2
=13/6+1/2
=13/6
=8/3
We now have the complete terms of the harmonic series, 3/2+6/7+3/5+6/13+3/8+6/19. Sn= 3/2+6/7+3/5+6/13+3/8+6/19 =3(1/2+2/7+1/5+2/13+1/8+2/19) = 94737 / 69160
3. If a and d are first term and common difference of Arithmetic Sequence respectively, then nth term of corresponding Harmonic sequence is: A an = a+(n-1)d B an = 1/a+(n-1)d C an = a/1+(n-1)d D an = a/a+(n-1)d Answer B
