Examples of geometric sequences. Here are a few examples of geometric sequences. Have a look!!

### Geometric sequence

Well, it’s an old topic from high school. So I’ll not go into much detail.

Suppose I have a sequence like .

Here is the first term and is the common ratio in the sequence.

Then this sequence is a geometric sequence.

There are also other sequences like arithmetic sequence, harmonic sequence and so on.

Now I’ll give some examples of geometric sequences.

#### Examples of geometric sequences

Here are some examples of geometric sequences.

*Disclaimer: None of these examples is mine. I have chosen these from some book or books. The references are at the end of the post.*

##### Example 1

According to Stroud and Booth (2013)* “Five numbers are in a geometric sequence. The first is 10 and the fifth is 160, what are the other three numbers?”

##### Solution

First of all, I’ll choose the sequence.

Let the geometric sequence be .

Here is the first term and is the common ratio in the sequence.

Now I already know that the first term is 10.

(1)

Also, I know that the fifth term is 160.

This means

From equation (1), I know that

Thus I can say that

(2)

Now from this example, I already know the first and the fifth terms of the sequence.

And, now I have to find out the other three terms.

For that, I’ll use both the equations (1) and (2).

So this gives the second term of the sequence as

Similarly, the third term of the sequence will be

In the same way, the fourth term of the sequence will be

Hence I can conclude that this the answer to this example.

Now comes the next example.

##### Example 2

According to Stroud and Booth (2013)* “The first, third and sixth terms of an arithmetic sequence form three successive terms of a geometric sequence. If the first term of both the arithmetic and geometric sequence is 8, find the second, third and fourth terms and the general term of the geometric sequence.”

##### Solution

Here I already know that the first, third and sixth terms of an arithmetic sequence form three successive terms of a geometric sequence.

So I’ll start with the arithmetic sequence.

###### Step 1

Let the arithmetic sequence be .

Now is the first term and is the common difference of the sequence.

So the first term of the sequence will be .

Similarly, the third term of the sequence will be .

And, the sixth term of the sequence will be .

So these three terms are in geometric sequence.

Now this means the common ratio of the geometric sequence is or .

Thus I can say

Here I already know that the first term of both the arithmetic and geometric sequence is 8. So this means .

Therefore I’ll substitute this value of in the equation to get

(3)

Next, I’ll get the value of .

###### Step 2

So I’ll simplify the equation (3) to get the value of as

Since doesn’t help me in this case, I’ll choose .

Thus with , the two terms and will be

and

So the common ratio of the geomteric sequence is

Hence the geometric sequence will be

Now the second, third and fourth terms of geometric sequence will be

and

Thus I can say that the second, third and fourth terms and the general term of the geometric sequence are and respectively.

Hence I can conclude that this is the answer to the given example.

Dear friends, this is the end of my today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

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