THE PROGRESSIONS. 315
387. To find the harmonic mean between two given quantities.
Let a, b be the two quantities, H their harmonic mean; then 1a, 1H, 1b are in A. P.;
1 H - 1 a = 1 b - 1 H, 2 H = 1a + 1 b, H = 2ab a + b.
388. Relation between the Arithmetic, Geometric, and Harmonic Means. If A, G, H be the arithmetic, geometric, and harmonic means between a and h, we have proved
A = a + b 2 (1). G = ab (2). H = 2ab a + b (3).
Therefore AH = a + b 2 2ab a + b = ab = G2,
that is, G is the geometric mean between A and H.
389. Miscellaneous Questions in the Progressions. Miscellaneous questions in the Progressions afford scope for much skill and ingenuity, the solution being often very neatly effected by some special artifice. The student will find the following hints useful.
1. If the same quantity be added to, or subtracted from, all the terms of an A. P., the resulting terms will form an A. P., with the same common difference as before. [Art. 365.]
