Properties of Common Ratio

 number sequence is a set or series of numbers that follow a specific pattern or a rule as it progresses. Such a sequence is a geometric sequence if the rule it follows is to multiply or divide the following number in the series each time. Thus, this number multiplied or divided at each stage of the series is called the common ratio.

Meaning of Common Ratio

A geometric progression or geometric sequence is a sequence in which each element is followed by another one obtained after multiplying or dividing a common number. The common ratio “r” is the common number derived after dividing two successive terms in the sequence. The common ratio formula helps calculate r for a given geometric progression.

For example, the sequence: 2, 6, 18, 54, 162, 486 is a geometric sequence.

In this, 2 x 3 = 6; 6 x 3 = 18; 18 x 3 = 54 and so on. Thus every number is succeeded by triple its value.

Formula

A geometric progression is in the form,

a1, (a1r), (a1r2), (a1r3), (a1r4)…

Here a1 represents the first term of the geometric sequence. Similarly, a1r is the second term, and r represents the common ratio. Thus, the common ratio formula of a geometric progression is

r = an / an – 1

Common Ratio r: Examples

Example 1

Using the formula of common ratio, find the common ratio in the given geometric progression.

1, 3, 9, 27, 81, 243 …

Answer:

The formula to find r is

r = an / an – 1

Thus a2 is divided by a1, a3 by a2 and so on. By dividing each number by the one preceding it, we can find r.

31 = 93 = 279 = 8127 = 3

Thus the geometric progression has the common multiple 3.

Therefore, r in the given geometric progression 1, 3, 9, 27, 81, 243, … = 3.

Example 2

Find the common ratio for a geometric progression where the nth term is given by the function an = 2(3)n-1.

Answer:

The function of the geometric sequence is an = 2(3)n-1.

Thus by putting the value of n = 1, we get

a1 = 2(3)1-1 = 2

Similarly, a2 = 2(3)2-1 = 6

Thus the series comes out to be 2, 6, 18, 54, 162, …

We know the formula to find r is

r = an / an – 1

Thus, r = 6 / 2 = 3.

We can check it with two other consecutive terms as well:

r = 162 / 154 = 3

Thus, r for the geometric progression where the nth term is given by the function an = 2(3)n-1 is equal to 3.

Properties of Common Ratio

Property 1

If every term of the geometric progression is multiplied or divided by a non zero number, then the resulting terms are also in a geometric progression. Furthermore, the resulting geometric progression also has the same common ratio.

Thus if a1, a2, a3, a3, a4, …  represents a geometric progression,

r = an / an – 1  for all n belonging to the set of all Natural Numbers,

and there is a non-zero constant, say k,

Then multiplying all terms of the geometric progression with k results in the geometric progression,

ka1, ka2, ka3, ka3, ka4, …

Thus, r for the new geometric progression will be

r = kan / kan – 1  which is equivalent to r = an / an – 1

For example, suppose there is a geometric progression 1, 2, 4, 8, 16, … with the common ratio 2.

If all the terms in this geometric progression are multiplied by 3, then the resulting geometric progression, i.e., 3, 6, 12, 24, 48, …, is also a geometric progression. Thus, r of this geometric progression is also 2.

Property 2

The reciprocals of all the terms in a geometric progression also result in the formation of a geometric progression.

Thus, if a1, a2, a3, a3, a4, … represents a geometric progression,

and r = an / an – 1  for all n belonging to the set of all Natural Numbers,

The series formed after taking the reciprocal of each term, i.e.,

1a1, 1a2, 1a3, 1a4, 1a5, … is also a geometric progression.

Furthermore, r of this resulting geometric progression will be equal to 1r.

For example, there is a sequence: 1, 2, 4, 8, 16, …

Thus, the reciprocals of the terms will be

11, 12, 14, 18, 116, …

Thus, r of the new sequence will be

r = an / an – 1

r = 1/21/1 = 12

Property 3

When all the elements in a geometric progression have the power of the same value, the resulting series also forms a geometric progression.

Thus, if a1, a2, a3, a3, a4, …  represents a geometric progression,

and r = an / an – 1  for all n belonging to the set of all Natural Numbers,

The series formed after raising all the terms to the power of the same value results in

a1k, a2k, a3k, a4k, a5k, …

Here, r = a2k / a1k = rk.

For example, there is a sequence: 1, 2, 4, 8, 16, … where the common ratio is 2.

Then, raising all the terms to the power 2,

1, 4, 16, 64, 256, …

Here, r = 41 = 4

The new r is 22 = 4.

Property 4

Three non-zero numbers a, b, c are in a geometric progression only if b2 = ac.

Property 5

The resulting series also constitutes a geometric progression when the terms of a geometric sequence are selected at regular intervals.

For example, there is a sequence: 1, 2, 4, 8, 16, 32, 64, 128, 256, …

Thus, if the terms of the sequence are selected alternatively, i.e.,

1, 4, 16, 64, 256, …

Then the resulting sequence is also in the form of a geometric progression.

Property 6

The product of the first and last terms of a geometric progression equals the product of terms equidistant from the start and the end of a finite sequence.

For example, there is a finite sequence: 1, 2, 4, 8, 16, 32.

Then the product of the first and last term of the sequence, i.e., 1 x 32, is equal to the terms equidistant from the first and last terms. For instance, the distance between the first and second term is equal to the distance between the second-last and last term. Thus, 1 x 32 = 2 x 16.

Property 7

The logarithm of all the terms in a geometric progression of non-zero positive terms, the resulting sequence is an arithmetic progression and vice-versa.

Conclusion

A geometric progression or geometric sequence is a sequence in which each element is followed by another one obtained after multiplying or dividing a common number.This common number is also called a common ratio and denoted by ‘r’.In the article we looked some properties of ‘r’.Hope you liked the article.