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Geometric sequences

Geometric sequences have a common ratio. To find the next term of a geometric sequence we multiply the current term by the common ratio.
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MICHAEL ANDERSON: Geometric sequences are similar to arithmetic sequences. But instead of there being a common difference, the amount that is added on each time, there is a common ratio– a common amount we multiply by to get from one term to the next. A simple geometric sequence is the powers of 2. In this case, g equals 2, 4, 8, 16, and 32, and so on. Just like arithmetic sequences are based upon the times tables, geometric sequences are based upon power sequences. G is equal to 2 to the power of 1, 2 to the power 2, 2 to power 3, 2 to the power 4, 2 to the power 5, and so on. The first term is 2.
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And the common ratio– the number that we multiply the previous term by to get the next term is also 2. We can express the term-to-term rule as G1, the first term being equal to 2. And to get the next term, we multiply the current term by 2. The position to term rule is 2 to the power of n. We can find the common ratio in a geometric sequence by dividing the current term by the previous term. That will give us r, the common ratio. In a geometric sequence, we can find the common ratio r by dividing the current term by the previous term. An example of a similar geometric sequence is H.
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H is equal to 6, 12, 24, 48, 96, and so on. This is still a geometric sequence. We can find the common ratio by dividing one term by the previous term. For example, 24 divided by 12 equals 2. Or 96 divided by 48 also gives us 2. The common ratio r is equal to 2. We can describe the term-to-term rule like this. H 1, our first term is 6. To get the next term, we multiply the current term by 2.
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Let’s have a look at the position to term rule. H is based on the number sequence, 2 to the power n. In this case, the sequence is always 3 times the value of 2n. So H n is equal to 3 multiplied by 2 to the power n. Let’s look at another example. J is equal to 4, 8, 16, 32, 64, and so on. We can describe the sequence using a term-to-term rule. J1, the first term, is 4. And we get the next term by multiplying the previous term by 2. We can write the position to term rule as Jn is equal to 2 to the power of n plus 1.
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The position to term rule can be found by multiplying 2 by 2 to the power of n. Another way of writing this is 2 to the power of n plus 1. Have a go of writing the first five terms of the following sequences– 3 to the power of n, 4 multiplied by 3 to the power of n, and 3 to the power of n plus 2.
Geometric sequences have a common ratio. To find the next term of a geometric sequence we multiply the current term by the common ratio.
We have seen that arithmetic sequences are based upon the times tables. Geometric sequences are based upon powers. If a geometric sequence has a term to term rule based upon multiplying by 2 then the position to term rule will be based on powers of 2.
In this video Michael explains how to find the common ratio of a geometric sequence, how to describe geometric sequences using a term to term rule and how to find the position to term rule. Michael concludes by setting a challenge.

Challenge

Write the first five terms for the following sequences:
a) \(3^n\)
b) \(4 \times 3^n\)
c) \(3^{(n + 2)}\)
Post your solutions to the challenge below.

Problem worksheet

Now complete questions 6 and 7 from this week’s worksheet
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