318 ALGEBRA.
17. If G be the geometric mean between two quantities A and B, show that the ratio of the arithmetic and harmonic means of A and G is equal to the ratio of the arithmetic and harmonic means of G and B. 18. To each of three consecutive terms of a G. P., the second of the three is added. Show that the three resulting quantities are in H. P.
Sum the following series:
19. 1 + 1 34 + 3 1 16 + ... to 6 terms. 20. 1 + 1 34 + 2 12 + ... to 6 terms. 21. (2 a + x) + 3 a + (4a - x) + ... to p terms. 22. 1 45 - 1 15 + 4 5 to 8 terms. 23. 1 45 + 1 15 + 3 5 + ... to 12 terms. 24. If x - a, y - b, and z - a be in G.P., prove that 2(y — a) is the harmonic mean between y - x and y - z. 25. If a, b, c, d be in A. P., a, e, f, d in G. P., a, g, h, d in H. P. respectively; prove that ad = ef =bh = eg. 26. If a2, b2, c2 be in A. P., prove that b + c, c + a, a + b are in H. P.
