The General Formula for a Sequence with a Common Ratio

Oct 13, 2025 by magicalceylontours.com 22 views

Understanding the Problem

When dealing with sequences, it's essential to understand the concept of a common ratio and how it affects the sequence's behavior. A common ratio is a value that is multiplied by each term in the sequence to get the next term. In this case, the common ratio is given as $\frac{3}{2}$ . Additionally, we are given that $f(5)=81$ , which provides us with a specific value of the sequence at a particular term.

The General Formula for a Sequence with a Common Ratio

The general formula for a sequence with a common ratio $r$ is given by:

f(x) = a \cdot r^{x-1}

where $a$ is the first term of the sequence, and $r$ is the common ratio.

Finding the First Term of the Sequence

To find the first term of the sequence, we can use the given value of $f(5)=81$ and the common ratio $r=\frac{3}{2}$ . Substituting these values into the general formula, we get:

81 = a \cdot \left(\frac{3}{2}\right)^{5-1}

Simplifying the equation, we get:

81 = a \cdot \left(\frac{3}{2}\right)^4

Solving for the First Term

To solve for the first term $a$ , we can divide both sides of the equation by $\left(\frac{3}{2}\right)^4$ :

a = \frac{81}{\left(\frac{3}{2}\right)^4}

Simplifying the expression, we get:

a = \frac{81}{\frac{81}{16}}

a = 16

Finding the Explicit Formula

Now that we have found the first term $a=16$ , we can substitute this value into the general formula to get the explicit formula for the sequence:

f(x) = 16 \cdot \left(\frac{3}{2}\right)^{x-1}

Conclusion

In conclusion, the explicit formula that represents the sequence is $f(x)=16\left(\frac{3}{2}\right)^{x-1}$ .

Comparison with Other Options

Let's compare our explicit formula with the other options given:

Option A: $f(x)=24\left(\frac{3}{2}\right)^{x-1}$
Option B: $f(x)=16\left(\frac{3}{2}\right)^{x-1}$
Option C: (not provided)

Our explicit formula matches option B, which is the correct answer.

Final Thoughts

In this problem, we used the concept of a common ratio and the general formula for a sequence to find the explicit formula for the sequence. We also used the given value of $f(5)=81$ to find the first term of the sequence. This problem demonstrates the importance of understanding the properties of sequences and how to use them to find explicit formulas.

Key Takeaways

The general formula for a sequence with a common ratio $r$ is given by $f(x) = a \cdot r^{x-1}$ .
To find the first term of the sequence, we can use the given value of the sequence at a particular term and the common ratio.
The explicit formula for the sequence can be found by substituting the first term into the general formula.

References

[1] "Sequences and Series". Khan Academy.
[2] "Mathematics for Computer Science". MIT OpenCourseWare.

Understanding the Problem

Q&A

Q: What is the general formula for a sequence with a common ratio?

A: The general formula for a sequence with a common ratio $r$ is given by:

f(x) = a \cdot r^{x-1}

where $a$ is the first term of the sequence, and $r$ is the common ratio.

Q: How do I find the first term of the sequence?

A: To find the first term of the sequence, you can use the given value of the sequence at a particular term and the common ratio. For example, if we are given that $f(5)=81$ and the common ratio is $\frac{3}{2}$ , we can substitute these values into the general formula to find the first term.

Q: What is the explicit formula for the sequence?

A: The explicit formula for the sequence is given by:

f(x) = 16 \cdot \left(\frac{3}{2}\right)^{x-1}

Q: How do I compare the explicit formula with other options?

A: To compare the explicit formula with other options, you can substitute the values of $x$ into the formula and see which one matches the given sequence. For example, if we substitute $x=5$ into the formula, we get:

f(5) = 16 \cdot \left(\frac{3}{2}\right)^{5-1}

Simplifying the expression, we get:

f(5) = 16 \cdot \left(\frac{3}{2}\right)^4

f(5) = 81

This shows that the explicit formula matches the given sequence.

Q: What are some key takeaways from this problem?

A: Some key takeaways from this problem are:

The general formula for a sequence with a common ratio $r$ is given by $f(x) = a \cdot r^{x-1}$ .
To find the first term of the sequence, you can use the given value of the sequence at a particular term and the common ratio.
The explicit formula for the sequence can be found by substituting the first term into the general formula.

Q: What are some related topics to this problem?

A: Some related topics to this problem are:

Sequences and Series
Common Ratios
Explicit Formulas

Q: What are some references for further learning?

A: Some references for further learning are:

[1] "Sequences and Series". Khan Academy.
[2] "Mathematics for Computer Science". MIT OpenCourseWare.

Conclusion

In conclusion, this problem demonstrates the importance of understanding the properties of sequences and how to use them to find explicit formulas. By using the general formula for a sequence with a common ratio, we can find the explicit formula for the sequence and compare it with other options.

Final Thoughts

This problem is a great example of how to use the general formula for a sequence with a common ratio to find the explicit formula for the sequence. By following the steps outlined in this problem, you can find the explicit formula for any sequence with a common ratio.

Key Takeaways

The general formula for a sequence with a common ratio $r$ is given by $f(x) = a \cdot r^{x-1}$ .
To find the first term of the sequence, you can use the given value of the sequence at a particular term and the common ratio.
The explicit formula for the sequence can be found by substituting the first term into the general formula.

References

[1] "Sequences and Series". Khan Academy.
[2] "Mathematics for Computer Science". MIT OpenCourseWare.