What are Geometric Progressions?

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Consider the following progression:

      \[1,\,\,2,\,\,4,\,\,8,\,\,16,\,\,32,\,\,...\]

Each term in this progression is twice the previous term. In other words, the ratio of every term to the previous term is fixed, and equal to 2.

Any progression in which the ratio of adjacent terms is fixed, is a Geometric Progression, or GP.

As in the case of APs, two parameters are sufficient to define a GP completely:

  1. First term: This is the first number of the sequence. In the series above, the first term is 1. For an arbitrary GP, the first term is generally denoted by a.

  2. Common ratio: This is the ratio of any term to the previous term, and is in general denoted by r. In the series above, r is equal to 2.

The table below lists some more examples of GPs, with their first terms and common ratios:

        \({2,\,\,\,1,\,\,\frac{1}{2},\,\,\frac{1}             {4},\,\,\frac{1}{8},...}\)           \({a = 2,\,\,r = \frac{1}{2}}\)
        \({ - \frac{1}{2},\,\,\frac{1}{6},\,\, - \frac{1}{{18}},\,\,\frac{1}{{54}},...}\)            \({a = - \frac{1}{2},\,\,r = - \frac{1}{3}}\)
        \({1,\,\,\pi ,\,\,{\pi ^2},\,\,{\pi ^3},\,\,{\pi ^4},...}\)            \({a = 1,\,\,r = \pi }\)
          \({0.1,\,\,0.01,\,\,0.001,\,\,0.0001,...}\)            \({a = 0.1,\,\,r = 0.1}\)

We note that the terms of a GP can take on any non-zero real values. The common ratio can be any non-zero real number. If \(r = 1\), we have a constant GP, because each term is equal to the previous term.

In terms of a and r, the terms of a GP can be written as follows:

\[a,\,\,ar,\,\,a{r^2},\,\,a{r^3},\,\,a{r^4},...\]

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