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What is a recursive rule for the geometric sequence? 6, -18, 54, -162..

What is a recursive rule for the geometric sequence? 6, -18, 54, -162..-example-1
User Dacuna
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Explanation:

To find the recursive rule for a geometric sequence, we need to determine the common ratio, which is the constant ratio between any two consecutive terms.

In this case, let's consider the sequence 6, -18, 54, -162...

To find the common ratio, we can divide any term by its previous term. Let's take the second and first terms:

-18 / 6 = -3

We can see that the common ratio is -3.

Now, we can use this common ratio to write the recursive rule for the sequence. A recursive rule expresses each term in the sequence in terms of the previous term(s).

Let's denote the first term of the sequence as a₁, and the common ratio as r.

The recursive rule for a geometric sequence is:

aₙ = r * aₙ₋₁

In our case, the first term a₁ is 6, and the common ratio r is -3.

Therefore, the recursive rule for the given geometric sequence is:

aₙ = -3 * aₙ₋₁

This rule means that each term in the sequence is obtained by multiplying the previous term by -3. To find any term in the sequence, you would need to know the previous term and apply this rule.

For example, to find the 4th term (a₄), you would multiply the 3rd term (a₃) by -3:

a₄ = -3 * a₃ = -3 * 54 = -162

By applying the recursive rule, you can continue to find other terms in the sequence.

User Umber Ferrule
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