The formula for the nth term of a harmonic sequence is an = 1 / (a + (n - 1) d).
Understanding the Harmonic Sequence Formula
A harmonic sequence is a sequence of numbers whose reciprocals form an arithmetic sequence. This fundamental relationship is key to understanding its formula. While an arithmetic sequence has a constant difference between consecutive terms, a harmonic sequence's terms are the reciprocals of an arithmetic progression.
The formula for the nth term of a harmonic sequence is derived directly from the formula for the nth term of an arithmetic sequence. If 1/a₁, 1/a₂, 1/a₃, ... is an arithmetic sequence, then a₁, a₂, a₃, ... is a harmonic sequence.
The formula for the nth term of a harmonic sequence is:
an = 1 / (a + (n - 1) d)
Here's what each variable in the formula represents:
| Variable | Description |
|---|---|
| an | The nth term of the harmonic sequence that you want to find. |
| a | The first term of the corresponding arithmetic sequence. |
| n | The position of the term in the sequence (e.g., 1st, 2nd, 3rd, etc.). |
| d | The common difference of the corresponding arithmetic sequence. |
How it Works: An Example
Let's illustrate how to use the formula with an example. Suppose we have an arithmetic sequence: 3, 6, 9, 12, ...
-
Identify 'a' and 'd' for the arithmetic sequence:
- The first term (a) is 3.
- The common difference (d) is 6 - 3 = 3.
-
Form the corresponding harmonic sequence:
- The harmonic sequence would be the reciprocals: 1/3, 1/6, 1/9, 1/12, ...
-
Find the 4th term of the harmonic sequence using the formula:
- We want to find
a₄, son = 4. a₄ = 1 / (a + (n - 1) d)a₄ = 1 / (3 + (4 - 1) * 3)a₄ = 1 / (3 + (3 * 3))a₄ = 1 / (3 + 9)a₄ = 1 / 12
- We want to find
This matches the 4th term we derived by taking the reciprocal of the 4th term of the arithmetic sequence.
Connection to Arithmetic Progressions
The core characteristic of a harmonic sequence is its direct link to an arithmetic progression. If a sequence of non-zero numbers a₁, a₂, a₃, ..., a_n is a harmonic sequence, then the sequence of their reciprocals, 1/a₁, 1/a₂, 1/a₃, ..., 1/a_n, forms an arithmetic sequence.
This relationship is crucial because it allows us to analyze and solve problems related to harmonic sequences by first converting them into their simpler arithmetic counterparts. Once converted, standard arithmetic sequence formulas can be applied, and the results then inverted to find the terms of the harmonic sequence.
