Properties of a geometric progression:
(a) If each term of a given geometric progression is multiplied or divided by the same number, then the resulting progression is also a G.P.
If a1, a2, a3, a4, .......... be a G.P. with common ratio r, then
a1c, a2c, a3c, a4c, .......... and \(\frac{a_1}{c}\), \(\frac{a_2}{c}\),\(\frac{a_3}{c}\), \(\frac{a_4}{c}\), .....are also G.P.s with common ratio r. c is the constant number.
(b) The reciprocals of the terms of a G.P. also form a G.P.
If a1, a2, a3, a4, .......... be a G.P. with common ratio r, then \(\frac{1}{a_1}\),\(\frac{1}{a_2}\),\(\frac{1}{a_3}\),\(\frac{1}{a_4}\), ....is also a G.P. with common ratio \(\frac{1}{r}\).
(c) If each term of a G.P. be raised to the same power, then the resulting progression is also a G.P.
If a1, a2, a3, a4, .......... be a G.P. with common ratio r, then \(a^k_1\), \(a^k_2\), \(a^k_3\), \(a^k_4\), ..... is also a G.P. with common ratio r.
(d) If a1, a2, a3, a4, .......... and b1, b2, b3, b4, .... be two G.P.s with common ratio r1 and r2 respectively, then the progressions a1b1, a2b2, a3b3, .... is a G.P. with common ratio r1r2, and \(\frac{a_1}{b_1}\), \(\frac{a_2}{b_2}\), \(\frac{a_3}{b_3}\), \(\frac{a_4}{b_4}\), ....is a G.P. with common ratio \(\frac{r_1}{r_2}.\)
(e) If a1, a2, a3, .......... be a G.P. with common ratio r, such that each term of the progression is a positive number, then log a1, log a2, log a3, .... is an A.P. with common difference log r.
Conversely, if log a1, log a2, log a3, ..... are terms of an A.P., then a1, a2, a3, .... are terms of a G.P.
(f) The Arithmetic mean between two positive numbers is always greater than equal to their Geometric Mean. A.M. ≥ G.M.
(g) If A and G be the arithmetic and geometric mean respectively between two numbers, then the numbers are \(A\pm\sqrt{A^2-G^2}.\)
