Progressions: Properties of G.P.

Progressions

If \({a_1},{a_2},{a_3},{\rm{ }} \ldots ..\) are in G.P. having common ratio \(‘r’\), then \(\frac{1}{{{a_1}}},\frac{1}{{{a_2}}},\frac{1}{{{a_3}}},......\) are in G.P. having common ratio \(\frac{1}{r}\).
If \({a_1},{a_2},{a_3},\) ….. are in G.P. having common ratio \(r\), then \({a_1}^n,{a_2}^n,{a_3}^n,....{a_n}^n\)are in G.P. having common ratio \({r^n}\).
If \({a_1},{a_2},{a_3},\) …… and \({b_1},{b_2},{b_3},\) …… are two GPs having common ratio \({r_1}\) and \({r_2}\) respectively, then \({a_1}{b_1},{a_2}{b_2},{a_3}{b_3},\) …… are also in G.P. having common ratio\('{r_1}{r_2}'\).
If \({a_1},{a_2},{a_3},\) …….. are in G.P. having common ratio \(‘r’\), then log \({a_1}\), log \({a_2}\), log \({a_3}\), …… are in \(A.P.\) having common difference ‘log \(r\)’. the converse also holds good. Suppose log \({a_1}\), log \({a_2}\), log \({a_3}\), …….. are in \(A.P.\) Then \({a_1}\), \({a_2}\), \({a_3}\), …… will be in G.P.