Harmonic Progression (HP)
Harmonic Progression (HP)
A Harmonic Progression (HP) is a sequence of numbers formed by taking the reciprocals of an arithmetic progression (AP). In other words, if each term of an AP is taken as the denominator and 1 is taken as the numerator, the resulting sequence is an HP.
Understanding Harmonic Progression
To understand HP, let's first recall what an Arithmetic Progression is:
An Arithmetic Progression is a sequence of numbers in which each term after the first is obtained by adding a constant, called the common difference, to the previous term.
Formula for the nth term of an AP:
If $a$ is the first term and $d$ is the common difference, the nth term ($a_n$) of an AP is given by:
$$ a_n = a + (n - 1)d $$
Converting AP to HP:
If we have an AP with terms $a_1, a_2, a_3, ...$, the corresponding HP will have terms $\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, ...$.
Formula for the nth term of an HP:
If $H_n$ represents the nth term of an HP, and the HP is derived from an AP with first term $a$ and common difference $d$, then:
$$ H_n = \frac{1}{a + (n - 1)d} $$
Properties of Harmonic Progression
- The sequence of reciprocals of a harmonic progression is an arithmetic progression.
- There is no common difference in an HP as there is in an AP. Instead, there is a pattern in the reciprocals of the terms.
- The nth term of an HP can be found using the formula for the nth term of the corresponding AP.
Differences Between AP, GP, and HP
| Property | Arithmetic Progression (AP) | Geometric Progression (GP) | Harmonic Progression (HP) |
|---|---|---|---|
| Definition | A sequence where each term after the first is obtained by adding a constant. | A sequence where each term after the first is obtained by multiplying the previous term by a constant. | A sequence where the reciprocals of the terms form an arithmetic progression. |
| Common Difference/Product | Common difference ($d$) | Common ratio ($r$) | No common difference, but the reciprocals have a common difference. |
| nth Term Formula | $a_n = a + (n - 1)d$ | $g_n = ar^{n-1}$ | $H_n = \frac{1}{a + (n - 1)d}$ |
| Sum Formula | $S_n = \frac{n}{2}(2a + (n - 1)d)$ | $S_n = \frac{a(1 - r^n)}{1 - r}$, for $r \neq 1$ | No direct formula, but can be calculated using the sum of the corresponding AP. |
Examples of Harmonic Progression
Example 1: Identifying HP
Determine if the sequence $\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, ...$ is an HP.
Solution:
The reciprocals of the terms are $2, 4, 6, ...$, which form an AP with a common difference of $2$. Therefore, the original sequence is an HP.
Example 2: Finding the nth Term
Find the 5th term of the HP whose corresponding AP has the first term $a = 3$ and common difference $d = 2$.
Solution:
The nth term of the corresponding AP is given by $a_n = a + (n - 1)d$. For the 5th term, $n = 5$:
$$ a_5 = 3 + (5 - 1) \cdot 2 = 3 + 8 = 11 $$
The 5th term of the HP is the reciprocal of the 5th term of the AP:
$$ H_5 = \frac{1}{a_5} = \frac{1}{11} $$
Example 3: Sum of Terms in HP
Find the sum of the first 4 terms of the HP whose corresponding AP has the first term $a = 1$ and common difference $d = 1$.
Solution:
First, find the terms of the AP:
$$ a_1 = 1, a_2 = 2, a_3 = 3, a_4 = 4 $$
Now, find the reciprocals to get the HP:
$$ H_1 = \frac{1}{1}, H_2 = \frac{1}{2}, H_3 = \frac{1}{3}, H_4 = \frac{1}{4} $$
The sum of the first 4 terms of the HP is:
$$ S_4 = H_1 + H_2 + H_3 + H_4 = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{25}{12} $$
In conclusion, Harmonic Progression is a fascinating concept that emerges from the reciprocals of an Arithmetic Progression. Understanding HP requires a grasp of AP and the relationship between their terms. The examples provided illustrate how to work with HPs, including finding specific terms and sums of terms.