Geometric Sequences

The Geometric sequence concept is most important for all the learners to understand. It is a sequence in which every term is equal to the previous term times a constant, non-zero multiplier known as a common factor. In addition to this, these types of sequences can be finite as well as infinite. Geometric sequences have a fixed number of terms and always share a common ratio. We will learn how to identify geometric sequences and apply the formula for the geometric sequences to find out the following terms. Let us discuss the concept of geometric sequences and their properties below.

How do geometric sequences work effectively?

The geometric sequence is also called a geometric progression that works very effectively. A geometric sequence is described by its initial number a, the common factor r, and the number of terms S. The equivalent general form of the geometric sequence is:

a, ar, arᶾ,…

An illustration of the geometric sequence with initial number 2, common factor 2, and eight terms is 2, 4, 8, 16, 32, 64, 128, and 256.

We also observe these sequences in interest rates, population development, and physics. That is why it is essential to learn the concept of the formula of geometry.

How to solve the geometric sequence?

There are various approaches to finding out the unknown elements of the geometric sequences. The most significant step is to find out the common ratio shared by the sequence since, in most of the formula of geometry, r is essential.

You can use the recursive formula and explicit formula that enables us to find any type of term in a geometric sequence. For instance, the common ratio is 9, and then every term is nine times the previous terms. Do you know how to use the recursive and explicit formulas for geometric sequences?

Example: Utilizing the recursive formula for the geometric sequences.

Write a recursive formula for the below geometric sequences.

{ 6, 9, 13.5, 20.25…..}

Solution: The 1st term is provided as 6. The common ratio may find out by dividing the 2nd term by the 1st term, that is:

   r = 9/6 = 1.5 

Then substitute the common ratio into this recursive formula for the geometric sequences such as below,

an = ran-1

an =1.5 an-1 for n ≥ 2

a1 = 6

Explicit rule

Let us start to observe the common expressions for the terms for geometric sequences. The below example will help you to learn more about the explicit rule.

 a1 = a

a₂ = a . r

a₃ = ar . r

     = 

a₄ =  ar² . r

     =  ar³ 

To find out the nth term, you have to multiply the 1st term by the ratio raised to the (n-1) term, and it means if you continue the equation shown above and include an , then below is an explicit formula.

an = a₁ rn-1

This explicit rule is much more flexible than the recursive rule, so it is essential to keep this significant rule in mind.

Properties of geometric sequences

Geometric sequences have loads of unique properties that are very helpful and useful. This geometric mean of 2 numbers is the square root of their product; for instance, the geometric mean of 20 and 20 is 20 because the product 20 × 20 = 400, and the square root of 400 is 20.

In these geometric sequences, every term is the geometric mean of the term before or after it, for instance, 2,4,8,16,32 in which 4 is the geometric mean of 2 and 8, 16 is the geometric mean of 8 and 32.

In addition to this, another type of property of geometric sequences depends upon the common factor; if the common factor is more than 1, then infinite geometric sequences would approach positive infinity. But if r is between 0 or 1, then geometric sequences will approach 0.

If r is less than -1, then the term tends towards both negative or positive infinity because they alternate between positive and negative values. The geometry formula plays a significant role in biology, physics, and other fields.

Conclusion

The geometric sequences are very helpful in mathematical models of scientific models of real-world processes. Moreover, the use of these particular sequences may assist with the study of populations that develop at a fixed amount over provided time and investment that earn interest. So that this recursive or general formula of geometry makes it possible to find the accurate values in upcoming days based upon the initial point or common factor. That is why it is significant to know about all the concepts of geometric sequences, as these sequences play a significant role in every learner’s life.