Enter The Correct Answer In The Box.The Explicit Formula For A Certain Geometric Sequence Is $f(n)=1,250(11)^{n-1}$. What Is The Exponential Function For The Sequence? Write Your Answer In The Form Shown.$f(n)=\frac{\square}{\square}$

Feb 27, 2025 by ADMIN 235 views

**Geometric Sequences and Exponential Functions**

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. The explicit formula for a geometric sequence is given by $f(n) = a \cdot r^{n-1}$ , where $a$ is the first term and $r$ is the common ratio.

The Explicit Formula for a Certain Geometric Sequence

The explicit formula for a certain geometric sequence is given by $f(n) = 1,250(11)^{n-1}$ . This formula indicates that the first term of the sequence is $1,250$ and the common ratio is $11$ .

Finding the Exponential Function for the Sequence

To find the exponential function for the sequence, we need to rewrite the explicit formula in the form $f(n) = \frac{\square}{\square}$ . This can be done by using the fact that $a \cdot r^{n-1} = \frac{a \cdot r^{n-1}}{1}$ .

Rewriting the Explicit Formula

We can rewrite the explicit formula as follows:

$f(n) = 1,250(11)^{n-1}$ $f(n) = \frac{1,250(11)^{n-1}}{1}$

Simplifying the Exponential Function

To simplify the exponential function, we can use the fact that $a \cdot r^{n-1} = a \cdot r^{n-1} \cdot \frac{1}{r^{n-1}}$ . This gives us:

$f(n) = \frac{1,250(11)^{n-1}}{1}$ $f(n) = \frac{1,250(11)^{n-1} \cdot \frac{1}{(11)^{n-1}}}{\frac{1}{(11)^{n-1}}}$ $f(n) = \frac{1,250 \cdot 1}{\frac{1}{(11)^{n-1}}}$ $f(n) = \frac{1,250}{\frac{1}{(11)^{n-1}}}$

Simplifying the Fraction

To simplify the fraction, we can use the fact that $\frac{a}{\frac{1}{b}} = a \cdot b$ . This gives us:

$f(n) = \frac{1,250}{\frac{1}{(11)^{n-1}}}$ $f(n) = 1,250 \cdot (11)^{n-1}$

The Exponential Function for the Sequence

The exponential function for the sequence is given by $f(n) = \frac{1,250}{(11)^{1-n}}$ . This is the final answer.

Conclusion

In this article, we have discussed geometric sequences and exponential functions. We have also found the exponential function for a certain geometric sequence using the explicit formula. The exponential function for the sequence is given by $f(n) = \frac{1,250}{(11)^{1-n}}$ .

Key Takeaways

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.
The explicit formula for a geometric sequence is given by $f(n) = a \cdot r^{n-1}$ , where $a$ is the first term and $r$ is the common ratio.
To find the exponential function for a sequence, we need to rewrite the explicit formula in the form $f(n) = \frac{\square}{\square}$ .
The exponential function for a sequence can be simplified by using the fact that $a \cdot r^{n-1} = a \cdot r^{n-1} \cdot \frac{1}{r^{n-1}}$ .

References

[1] "Geometric Sequences and Series." Math Open Reference, mathopenref.com/geomseq.html.
[2] "Exponential Functions." Math Is Fun, mathisfun.com/algebra/exponential-functions.html.

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 explicit formula for a geometric sequence? The explicit formula for a geometric sequence is given by $f(n) = a \cdot r^{n-1}$ , where $a$ is the first term and $r$ is the common ratio.
How do I find the exponential function for a sequence? To find the exponential function for a sequence, we need to rewrite the explicit formula in the form $f(n) = \frac{\square}{\square}$ .

Glossary

Geometric sequence: 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.
Explicit formula: The formula that gives the nth term of a sequence.
Exponential function: A function of the form $f(n) = a \cdot r^{n-1}$ , where $a$ is the first term and $r$ is the common ratio.
Common ratio: The fixed, non-zero number that is multiplied by the previous term to get the next term in a geometric sequence.
Geometric Sequences and Exponential Functions: Q&A =====================================================

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 explicit formula for a geometric sequence?

A: The explicit formula for a geometric sequence is given by $f(n) = a \cdot r^{n-1}$ , where $a$ is the first term and $r$ is the common ratio.

Q: How do I find the exponential function for a sequence?

A: To find the exponential function for a sequence, we need to rewrite the explicit formula in the form $f(n) = \frac{\square}{\square}$ .

Q: What is the exponential function for a sequence?

A: The exponential function for a sequence is given by $f(n) = \frac{a}{r^{1-n}}$ .

Q: How do I simplify the exponential function for a sequence?

A: To simplify the exponential function for a sequence, we can use the fact that $a \cdot r^{n-1} = a \cdot r^{n-1} \cdot \frac{1}{r^{n-1}}$ . This gives us:

$f(n) = \frac{a}{r^{1-n}}$ $f(n) = \frac{a \cdot r^{n-1} \cdot \frac{1}{r^{n-1}}}{\frac{1}{r^{n-1}}}$ $f(n) = \frac{a \cdot 1}{\frac{1}{r^{n-1}}}$ $f(n) = a \cdot r^{1-n}$

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 the previous 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, we need to divide the second term by the first term.

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, we need to look at the explicit formula for the sequence.

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

A: The nth term in a geometric sequence is the term that is in the nth position in the sequence.

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

A: To find the nth term in a geometric sequence, we need to use the explicit formula for the sequence.

Q: What is the sum of a geometric sequence?

A: The sum of a geometric sequence is the sum of all the terms in the sequence.

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

A: To find the sum of a geometric sequence, we need to use the formula for the sum of a geometric sequence.

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:

$S_n = \frac{a(1-r^n)}{1-r}$

Q: How do I use the formula for the sum of a geometric sequence?

A: To use the formula for the sum of a geometric sequence, we need to plug in the values of $a$ , $r$ , and $n$ into the formula.

Q: What is the difference between a geometric sequence and an arithmetic 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. An arithmetic sequence is a type of sequence where each term after the first is found by adding a fixed, non-zero number called the common difference to the previous term.

Q: How do I determine whether a sequence is geometric or arithmetic?

A: To determine whether a sequence is geometric or arithmetic, we need to look at the relationship between the terms in the sequence.

Q: What is the importance of geometric sequences and exponential functions?

A: Geometric sequences and exponential functions are important in many areas of mathematics and science, including finance, economics, and physics.

Q: How do I apply geometric sequences and exponential functions in real-life situations?

A: To apply geometric sequences and exponential functions in real-life situations, we need to use the formulas and concepts that we have learned in this article.

Q: What are some common applications of geometric sequences and exponential functions?

A: Some common applications of geometric sequences and exponential functions include:

Finance: Geometric sequences and exponential functions are used to calculate interest rates and investment returns.
Economics: Geometric sequences and exponential functions are used to model economic growth and inflation.
Physics: Geometric sequences and exponential functions are used to describe the behavior of physical systems, such as population growth and radioactive decay.

Q: How do I use geometric sequences and exponential functions to solve problems?

A: To use geometric sequences and exponential functions to solve problems, we need to apply the formulas and concepts that we have learned in this article.

Q: What are some common mistakes to avoid when working with geometric sequences and exponential functions?

A: Some common mistakes to avoid when working with geometric sequences and exponential functions include:

Not using the correct formula for the sum of a geometric sequence.
Not plugging in the correct values into the formula for the sum of a geometric sequence.
Not using the correct formula for the nth term of a geometric sequence.
Not plugging in the correct values into the formula for the nth term of a geometric sequence.

Q: How do I troubleshoot common mistakes when working with geometric sequences and exponential functions?

A: To troubleshoot common mistakes when working with geometric sequences and exponential functions, we need to:

Review the formulas and concepts that we have learned in this article.
Check our work for errors.
Use online resources and calculators to check our work.
Ask for help from a teacher or tutor.