Harmonic Progression Sum
An Arithmetic Progression (AP), also known as an Arithmetic sequence, is a set of numbers characterized by a constant difference between successive terms. On the other hand, a Harmonic Progression (HP) or Harmonic Sequence is generated by taking the reciprocals of an Arithmetic Progression.
Creating a Harmonic Progression or 1/AP is a straightforward process. Using the formula for the nth term in an Arithmetic Progression, a + (n-1)d, we can quickly generate the Harmonic Progression sequence.
However, calculating the sum of this progression can be tedious. One approach involves either iterating through the sequence or employing approximations to create a formula that provides an accurate value up to a few decimal places.
To find the Sum of n terms in a Harmonic Progression (Sn) for the sequence 1/a, 1/a+d, 1/a+2d, …, 1/a+(n−1)d, the formula is:
Sn= 1/d.ln{2a+(2n−1)d}/(2a − d)}
where,
- “a” denotes the first term of the Harmonic Progression (H.P.)
- “d” represents the common difference in the Harmonic Progression (H.P.)
- “ln” stands for the natural logarithm.
Harmonic Progression
A Harmonic Progression (H.P.) is a mathematical sequence generated by taking the reciprocals of an Arithmetic Progression. In this sequence, each term is the harmonic mean of its adjacent terms, this series is called Harmonic Progression.
A Harmonic Progression of separate unit fractions cannot add to an integer (unless in the specific case where (a = 1 and d = 0). The reason lies in the fact that the progression will contain at least one denominator divisible by a prime number that does not a divisor of any other denominator. Harmonic Progression is also called Harmonic Sequence.
In this article, we will discuss the definition, applications, and formula of Harmonic Progression, and understand the difference and relation between arithmetic mean, geometric mean, and harmonic mean to calculate Harmonic Progression in mathematics.
Table of Content
- What is Harmonic Progression (HP)?
- Harmonic Progression Example
- Harmonic Progression Formula
- Harmonic Progression Formula for nth Term
- Harmonic Progression Sum
- What is Harmonic Sequence?
- Harmonic Mean
- Arithmetic Mean (AM), Geometric Mean (GM) and Harmonic Mean (HM)
- Applications of Harmonic Progression
- Solved Examples on Harmonic Progression
- Harmonic Progression Questions