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2.11 F) Geometric Sequences
2.11 F) Geometric Sequences
A geometric sequence is where the next term is found by multiplying the previous term by a particular value that is always the same. This value can be positive or negative and large or small.
Example 1
Here is an example of a geometric series.
Here is an example of a geometric series.
By looking at this geometric series we can see that we multiply the previous term by two to obtain the next term in the sequence. For example, the first term in the sequence is 4. We multiply 4 by 2 to obtain the second term, which is 8. To obtain the third term, we multiply the second term (8) by 2, which means that the third term is 16. We can continue multiplying the previous term by 2 to obtain the next term in the sequence. A typical question in the exam might ask you to find the next few terms in a geometric sequence.
Let’s suppose that we are asked to find the next 3 terms in the geometric sequence above. We obtain the next term in the sequence by multiplying the previous term by 2. Therefore, we multiply 32 by 2, which gives us 64.
Let’s suppose that we are asked to find the next 3 terms in the geometric sequence above. We obtain the next term in the sequence by multiplying the previous term by 2. Therefore, we multiply 32 by 2, which gives us 64.
The next two terms can be found in a similar way; we multiply the previous terms by 2.
Therefore, the next 3 terms in the sequence are 64, 128 and 256.
Example 2
The sequence below is a geometric sequence. Find the next 3 terms in the sequence below. you may use a calculator.
The sequence below is a geometric sequence. Find the next 3 terms in the sequence below. you may use a calculator.
We are told that the above sequence is a geometric sequence, which means that we obtain the next term by multiplying the previous term by a value. For most of these questions, you should be able to see what the value is quite quickly, but if you are unsure, you can find what the value is by dividing two consecutive terms; we divide the later term by the earlier term. It does not matter which consecutive terms you use because any consecutive terms will give you the same value. I am going to use the first and second term; the first term is 2 and the second term is 6. We find the difference by dividing the later term by the earlier term. Therefore, we divide 6 by 2, which gives us 3. Therefore, the next term is obtained by multiplying the previous term by 3.
We are asked to find the next 3 terms and the working is shown below.
We are asked to find the next 3 terms and the working is shown below.
The next 3 terms are 162, 486 and 1,458.
Example 3
It may be the case that the terms in a geometric sequence become smaller. We still use the same process as before to find the next few terms.
Find the next 3 terms in the sequence below.
It may be the case that the terms in a geometric sequence become smaller. We still use the same process as before to find the next few terms.
Find the next 3 terms in the sequence below.
From the above sequence, you can see that the terms have been multiplied by ½ or divided by 2 (multiply by ½ and dividing by 2 are the same).
If you couldn’t see the what the value is, you could have obtained it by dividing two consecutive terms. I am going to use the first two consecutive terms. We find the value by dividing the later term by the earlier term. Therefore, we divide 80 by 160, which gives us a value of ½.
We now know that we obtain the next term in the sequence by multiplying the previous term by ½ (or dividing by 2). Therefore, we can find the next 3 terms.
If you couldn’t see the what the value is, you could have obtained it by dividing two consecutive terms. I am going to use the first two consecutive terms. We find the value by dividing the later term by the earlier term. Therefore, we divide 80 by 160, which gives us a value of ½.
We now know that we obtain the next term in the sequence by multiplying the previous term by ½ (or dividing by 2). Therefore, we can find the next 3 terms.
The next 3 terms are 10, 5 and 2.5.