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Understanding Geometric Sequences

A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This common ratio is denoted by the letter 'r'. The general form of a geometric sequence is given by:

a, ar, ar^2, ar^3, ...

where 'a' is the first term of the sequence.

Identifying the Recursive Rule

To identify the recursive rule for a geometric sequence, we need to find the common ratio 'r' and the first term 'a'. The recursive rule is given by:

an = ar^(n-1)

where 'an' is the nth term of the sequence.

Finding the Recursive Rule for the Given Sequence

The given sequence is:

6, -18, 54, -162, ...

We can see that each term is obtained by multiplying the previous term by -3. Therefore, the common ratio 'r' is -3.

Calculating the First Term

To calculate the first term 'a', we can use the fact that the first term is equal to the first term of the sequence. In this case, the first term is 6.

Writing the Recursive Rule

Now that we have found the common ratio 'r' and the first term 'a', we can write the recursive rule for the given sequence.

Recursive Rule

an = -3 * (n-1)

where 'an' is the nth term of the sequence.

Verifying the Recursive Rule

To verify the recursive rule, we can plug in the values of 'n' and calculate the corresponding terms of the sequence.

For n = 1, an = -3 * (1-1) = 6

For n = 2, an = -3 * (2-1) = -18

For n = 3, an = -3 * (3-1) = 54

For n = 4, an = -3 * (4-1) = -162

We can see that the recursive rule is correct.

Conclusion

In this article, we have discussed the concept of geometric sequences and how to identify the recursive rule for a given sequence. We have used the given sequence to find the common ratio 'r' and the first term 'a', and then wrote the recursive rule for the sequence. We have also verified the recursive rule by plugging in the values of 'n' and calculating the corresponding terms of the sequence.

Frequently Asked Questions

  • What is a geometric sequence? A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
  • What is the recursive rule for a geometric sequence? The recursive rule for a geometric sequence is given by: an = ar^(n-1)
  • How do I find the recursive rule for a given sequence? To find the recursive rule for a given sequence, you need to find the common ratio 'r' and the first term 'a'. You can then use the formula: an = ar^(n-1) to write the recursive rule.

Further Reading

  • Geometric Sequences: A Comprehensive Guide
  • Recursive Rules: A Guide to Understanding and Applying Them
  • Math Formulas: A Collection of Essential Formulas and Equations

References

Understanding Geometric Sequences

A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This common ratio is denoted by the letter 'r'. The general form of a geometric sequence is given by:

a, ar, ar^2, ar^3, ...

where 'a' is the first term of the sequence.

Q&A: Geometric Sequences

Q: What is a geometric sequence?

A: A geometric sequence is a type of sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Q: What is the common ratio in a geometric sequence?

A: The common ratio in a geometric sequence is the fixed, non-zero number that is multiplied by each term to get the next term.

Q: How do I find the common ratio in a geometric sequence?

A: To find the common ratio in a geometric sequence, you can divide any term by the previous term. For example, if the sequence is 2, 6, 18, 54, ..., you can divide 6 by 2 to get 3, which is the common ratio.

Q: What is the first term in a geometric sequence?

A: The first term in a geometric sequence is the first term of the sequence.

Q: How do I find the first term in a geometric sequence?

A: To find the first term in a geometric sequence, you can use the formula: a = an / r^(n-1), where 'a' is the first term, 'an' is the nth term, and 'r' is the common ratio.

Q: What is the recursive rule for a geometric sequence?

A: The recursive rule for a geometric sequence is given by: an = ar^(n-1), where 'an' is the nth term, 'a' is the first term, and 'r' is the common ratio.

Q: How do I find the recursive rule for a geometric sequence?

A: To find the recursive rule for a geometric sequence, you need to find the common ratio 'r' and the first term 'a'. You can then use the formula: an = ar^(n-1) to write the recursive rule.

Q: What is the formula for the nth term of a geometric sequence?

A: The formula for the nth term of a geometric sequence is given by: an = a * r^(n-1), where 'an' is the nth term, 'a' is the first term, and 'r' is the common ratio.

Q: How do I find the sum of a geometric sequence?

A: To find the sum of a geometric sequence, you can use the formula: S = a * (1 - r^n) / (1 - r), where 'S' is the sum, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.

Q: What is the formula for the sum of a geometric sequence?

A: The formula for the sum of a geometric sequence is given by: S = a * (1 - r^n) / (1 - r), where 'S' is the sum, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.

Conclusion

In this article, we have answered some of the most frequently asked questions about geometric sequences. We have covered topics such as the common ratio, the first term, the recursive rule, and the formula for the nth term and the sum of a geometric sequence. We hope that this article has been helpful in understanding geometric sequences.

Further Reading

  • Geometric Sequences: A Comprehensive Guide
  • Recursive Rules: A Guide to Understanding and Applying Them
  • Math Formulas: A Collection of Essential Formulas and Equations

References