Confirmed line item performance

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression. In other words, it is a sequence of the form

$1/a,\frac{1}{a+d},\frac{1}{a+2d},\frac{1}{a+3d},\cdots ,\frac{1}{a+kd},$

where −1/d is not a natural number and k is a natural number.

(Terms in the form $\frac{x}{y+z}$ can be expressed as $\frac{\frac{x}{y}}{\frac{y+z}{y}}$ , we can let $\frac{x}{y}=a$ and $\frac{z}{y}=kd$.)

Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms.

Examples

12, 6, 4, 3, $\frac{12}{5}$, 2, … , $\frac{12}{1+n}$

10, 30, −30, −10, −6, − $\frac{30}{7}$, … , $\frac{10}{1-\frac{2n}{3}}$

Use in geometry

If collinear points A, B, C, and D are such that D is the harmonic conjugate of C with respect to A and B, then the distances from any one of these points to the three remaining points form harmonic progression. Specifically, each of the sequences AC, AB, AD; BC, BA, BD; CA, CD, CB; and DA, DC, DB are harmonic progressions, where each of the distances is signed according to a fixed orientation of the line.

See also

• Geometric progression
• Harmonic series

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

• Mastering Technical Mathematics by Stan Gibilisco, Norman H. Crowhurst, (2007) p. 221
• Standard mathematical tables by Chemical Rubber Company (1974) p. 102
• Essentials of algebra for secondary schools by Webster Wells (1897) p. 307

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