In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression, which is also known as an arithmetic sequence. Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms. As a third equivalent characterization, it is an infinite sequence of the form
1 a , 1 a + d , 1 a + 2 d , 1 a + 3 d , ⋯ , {\displaystyle {\frac {1}{a}},\ {\frac {1}{a+d}},\ {\frac {1}{a+2d}},\ {\frac {1}{a+3d}},\cdots ,}
where a is not zero and −a/d is not a natural number, or a finite sequence of the form
1 a , 1 a + d , 1 a + 2 d , 1 a + 3 d , ⋯ , 1 a + k d , {\displaystyle {\frac {1}{a}},\ {\frac {1}{a+d}},\ {\frac {1}{a+2d}},\ {\frac {1}{a+3d}},\cdots ,\ {\frac {1}{a+kd}},}
where a is not zero, k is a natural number and −a/d is not a natural number or is greater than k.
Examples In the following n is a natural number, in sequence: n = 1 , 2 , 3 , 4 , … {\displaystyle \ n=1,\ 2,\ 3,\ 4,\ \ldots \ }
1 , 1 2 , 1 3 , 1 4 , 1 5 , 1 6 , … , 1 n , … {\displaystyle 1,{\tfrac {\ 1\ }{2}},\ {\tfrac {\ 1\ }{3}},\ {\tfrac {\ 1\ }{4}},\ {\tfrac {\ 1\ }{5}},\ {\tfrac {\ 1\ }{6}},\ \ldots \ ,\ {\tfrac {\ 1\ }{n}},\ \ldots \ } is called the harmonic sequence 12, 6, 4, 3, 12 5 , 2 , … , 12 n , … {\displaystyle \ {\tfrac {12}{\ 5\ }},\ 2,\ \ldots \ ,\ {\tfrac {12}{\ n\ }},\ \ldots \ }
30, −30, −10, −6, − 30 7 , … , 30 ( 3 − 2 n ) , … {\displaystyle \ -{\tfrac {30}{\ 7\ }},\ \ldots \ ,\ {\tfrac {30}{\ \left(3\ -\ 2n\right)\ }},\ \ldots \ }
10, 30, −30, −10, −6, … , 30 ( 5 − 2 n ) , … {\displaystyle \ \ldots \ ,\ {\tfrac {30}{\ \left(5\ -\ 2n\right)\ }},\ \ldots \ }
Sums of harmonic progressions
Infinite harmonic progressions are not summable (sum to infinity). It is not possible for a harmonic progression of distinct unit fractions (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.
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