A harmonic sequence, also known as a harmonic progression, is a sequence of numbers in which the difference between the reciprocals of any two consecutive terms is constant. In other words, a harmonic sequence is formed by taking the reciprocals of every term in an arithmetic sequence. The best-known harmonic sequence is 1, 1/2, 1/3, 1/4, ..., whose corresponding arithmetic sequence is simply the counting numbers 1, 2, 3, 4, ....
Here are some key properties of harmonic sequences:
- The general form of a harmonic sequence is $$\frac{1}{a}, \frac{1}{a+d}, \frac{1}{a+2d}, \frac{1}{a+3d}, ...$$, where $$a$$ is the first term and $$d$$ is the common difference.
- The $$n$$th term of a harmonic sequence can be expressed as $$\frac{1}{a+ (n-1)d}$$.
- The sum of the first $$n$$ terms of a harmonic sequence is given by the formula $$S_n = \frac{n}{2}\left(\frac{2a+(n-1)d}{a+(n-1)d}\right)$$.
- The sum of an infinite harmonic sequence is called a harmonic series, and it diverges, meaning it does not converge to a limit.
For example, consider the arithmetic sequence $$5, 10, 15, 20, 25, ...$$ with a common difference of $$5$$. Its corresponding harmonic sequence is $$1/5, 1/10, 1/15, 1/20, 1/25, ...$$.
