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### Geometric Progressions

A geometric progression is a sequence of numbers in which each successive number is obtained by multiplying a fixed quantity with the previous number.

The fixed or constant quantity that is multiplied to get the next term in a geometric progression is known as the common ratio. It is commonly represented by the letter r.

Let each term from the beginning in a geometric progression be represented by the following:
a1, a2, a3, a4, a5, ...
(where a1 is the first term, a2 the second term, and so on.)

In the above sequence, the second term (a2) is obtained by multiplying a fixed quantity, the common ratio, to the first term (a1). That is, if the common ratio is r,
a2 = a1 x r
Similarly the third term (a3) is obtained by multiplying a fixed quantity (the common ratio) to the second term (a2). That is, if the common ratio is r,
a3 = a2 x r
Thus the common ratio (r) of a geometric progression can be obtained by dividing its any term by the previous term. Thus, in the following geometric progression:
2, 4, 8, 16, ...
the common ratio is 4/2 = 2. Similarly the common ratios of the following geometric progressions are given:

• 3, 9, 27, ... (common ratio, r = 3)
• 2, 6, 18, ... (common ratio, r = 2)
• 1/2, 1/4, 1/8, 1/16, ... (common ratio, r = 1/2)

Thus, if in a geometric progression, a is the first term and r is the common ratio, then

• Second term = a x r = ar
• third  term = (second term) x r = ar2
• fourth term = (third term) x r = ar3
• and so on..

Thus a geometric progression with first term a and common ratio r can be represented by
ar, ar2, ar3, ar4, ...
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