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If each term following the first is derived by multiplying the preceding term by a constant quantity, a sequence of non-zero numbers is said to be in Geometric Progression (abbreviated as G.P.) (positive or negative). The constant ratio, also known as the Geometric Progression’s common ratio, is calculated by dividing each word by the term immediately before it.
At each stage of the sequence, we multiply or divide by the common ratio. By dividing two successive pairs of phrases, we may find it. It makes no difference which pair is chosen as long as they are next to one another.
Common Ratio
A geometric sequence’s common ratio is the distance between each number. The common ratio is the ratio of two consecutive numbers, i.e. a number divided by the previous number in the series.
The term at which a certain series or sequence line arithmetic progression or geometric progression finishes or terminates is known as the last term. The first term of a series or any sequence, such as arithmetic progression, geometric progression, harmonic progression, and so on, is commonly symbolized with a little ‘l’. It’s usually written with a small ‘a’. the total number of words in a specific series is represented by the letter ‘n’.
Formula for Common Ratio
A geometric progression is represented in general as a1, (a1r), (a1r²), (a1r³), (a1r4),…, where a1 is the first term of GP, a1r is the second term of GP, and r is the common ratio. As a result, the geometric progression’s common ratio formula is as follows:
Common ratio(r)= an / an-1
Where,
an = The geometric progression nth term
an -1 = The geometric progression (n – 1)th term
We will always get this value if you divide (that is, calculate the ratio of) successive terms. The common ratio formula is used to get the common ratio for a geometric progression.
Examples of Common Ratio
Example: 1
Using the common ratio formula, find the common ratio for the geometric sequence 1, 2, 4, 8, 16,…
Solution:
To see if a common ratio exists, divide each term by the previous term.
2 / 1= 4 / 2 = 8 / 4 = 16 / 8 = 2
Because there is a common multiple, 2, known as the common ratio, the sequence is geometric.
Answer: r = 2 is the common ratio.
Example: 2
48,12,4, 2,… Is this a geometric sequence? Find the common ratio if that’s the case.
Solution:
To establish whether a common ratio exists, divide each term by the previous term.
12 / 48 = 1 / 4 4 / 12= 1 / 3 2 / 4 = 1 / 2
Because there is no common ratio, the sequence is not geometric.
Conclusion
We conclude in this article that, in a geometric sequence, the common ratio is the distance between each number. The common ratio is named for the fact that it is the same for all numbers, or common, and it is also the ratio between two successive integers in a series. The common ratio of a geometric sequence or geometric series is the ratio of one phrase to the previous term. The variable r is commonly used to represent this ratio.
