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:
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,
Thus, if in a geometric progression, a is the first term and r is the common ratio, then
Thus a geometric progression with first term a and common ratio r can be represented by
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 rSimilarly 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 rThus 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, ...Related posts:
- Sum of 'n' terms of a geometric progression
- Sum of an infinite geometric progression
- Progressions in general
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