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Geometric Sequences

Writing and Using Explicit Rules for Geometric Sequences

Geometric sequences, just like arithmetic sequences, can be described using explicit rules. For geometric sequences, the first term, is also used to find the rule. Instead of the common difference of the arithmetic sequences, the common ratio is used for the geometric ones.
Rule

Explicit Rule of Geometric Sequences

All geometric sequences have a common ratio, Using the common ratio, together with the value of the first term of the sequence, an explicit rule describing the sequence can be found. By expressing the terms in a geometric sequence using and a pattern emerges. Note that is equal to , and that can be written as

Using and

When increases by the exponent on increases by as well. Due to this, and that the exponent is when is the exponent is always less than Expressing this in a general form gives the explicit rule.

Thus, knowing and is enough to write the explicit rule of a geometric sequence. Notice that the common ratio could be a negative number, leading to alternating positive and negative terms in the sequence.
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Exercise

The first four terms of a geometric sequence are Find the explicit rule describing the geometric sequence. Then, use the rule to find the eighth term of the sequence.

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Solution

To write the explicit rule for the sequence, we first have to find the common ratio, . To do so, we can divide any term in the sequence by the term that precedes it. Let's use the second and first term. Substituting and into the general rule for geometric sequences gives the desired rule. Now, we can find the eighth term in the sequence by substituting into the rule above.

The eighth term in the sequence is
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Exercise

Pelle's good friend, Lisa, decides to play a trick on Pelle. While he is away, she rearranges his pellets so that they are grouped in a geometric sequence instead of an arithmetic one. The first group has pellets, the second has the third has and so on. Find a rule describing this sequence. After finishing the seventh group, Lisa counted remaining pellets. Use the rule to figure out whether there are enough to make an eighth group.

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Solution

To begin, we'll write the explicit rule describing this particular geometric sequence. It is given that To find the common ratio, we can divide the second term by the first. To write the rule, we can substitute and into the general rule for geometric sequences. To find if there are enough pellets to finish the eighth group, we must know the eighth term in the sequence. We'll substitute into the rule.

The eighth term, is greater than the number of remaining pellets, Thus, there are not enough pellets for Lisa to make the eighth group.
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Exercise

For a geometric sequence, it is known that the common ratio is positive, and that Find the explicit rule for the sequence and give its first six terms.

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Solution

The terms we've been given are not consecutive. Therefore, we can't directly find However, the terms and are positions apart, so the ratio between them must be

This gives the equation which we can solve for

Now that we know the common ratio, we have to find as well, to be able to write the explicit rule. Knowing one term, a subsequent one can by found by multiplying by Therefore, a previous term is instead found by dividing by Using and this way, we can find With and we have enough information to state the explicit rule.

The desired explicit rule is We already know the terms and Let's use the rule to find the remaining three.

The terms and are evaluated similarly.

Thus, the first six terms of the sequence are

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