Find The Next Term In The Sequence: $5, 10, 20, 40, \ldots$A. 42 B. -80 C. 160 D. 80

Feb 27, 2025 by ADMIN 90 views

**Finding the Next Term in a Geometric Sequence**

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. In this problem, we are given a sequence of numbers: $5, 10, 20, 40, \ldots$ . Our task is to find the next term in this sequence.

Identifying the Common Ratio

To find the next term in the sequence, we need to identify the common ratio. We can do this by dividing each term by the previous term. Let's calculate the common ratio:

$\frac{10}{5} = 2$
$\frac{20}{10} = 2$
$\frac{40}{20} = 2$

As we can see, the common ratio is $2$ . This means that each term in the sequence is obtained by multiplying the previous term by $2$ .

Finding the Next Term

Now that we have identified the common ratio, we can find the next term in the sequence. We can do this by multiplying the last term by the common ratio:

$40 \times 2 = 80$

Therefore, the next term in the sequence is $80$ .

Conclusion

In this problem, we were given a geometric sequence and asked to find the next term. We identified the common ratio by dividing each term by the previous term and found that it was $2$ . We then used this common ratio to find the next term in the sequence, which was $80$ .

Answer

The correct answer is D. 80.

Why is this the correct answer?

This is the correct answer because we have identified the common ratio and used it to find the next term in the sequence. The common ratio is $2$ , and multiplying the last term by this ratio gives us the next term, which is $80$ .

What if the common ratio was not 2?

If the common ratio was not $2$ , we would not have been able to find the next term in the sequence. However, in this case, the common ratio is $2$ , and we have used it to find the next term.

What is the significance of this problem?

This problem is significant because it illustrates the concept of geometric sequences and how to find the next term in a sequence. Geometric sequences are used in many real-world applications, such as finance, economics, and engineering.

Real-World Applications of Geometric Sequences

Geometric sequences have many real-world applications, including:

Finance: Geometric sequences are used to calculate interest rates and investment returns.
Economics: Geometric sequences are used to model economic growth and inflation.
Engineering: Geometric sequences are used to design and optimize systems, such as electrical circuits and mechanical systems.

Conclusion

In conclusion, finding the next term in a geometric sequence involves identifying the common ratio and using it to find the next term. This problem illustrates the concept of geometric sequences and how to find the next term in a sequence. Geometric sequences have many real-world applications, including finance, economics, and engineering.

References

"Geometric Sequences" by Math Open Reference. Retrieved from https://www.mathopenref.com/sequencesgeometric.html
"Geometric Sequences and Series" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra2/x2f4f7f7/x2f4f7f8

Additional Resources

"Geometric Sequences" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/GeometricSequence.html
"Geometric Sequences and Series" by MIT OpenCourseWare. Retrieved from https://ocw.mit.edu/courses/mathematics/18-01sc-single-variable-calculus-fall-2010/unit-1-derivatives-and-differentiation/unit-1-1-basic-differentiation-rules/MIT18_01SC_unit1_1_1.html
Geometric Sequences 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: How do I find the common ratio of a geometric sequence?

A: To find the common ratio, you can divide each term by the previous term. For example, if the sequence is $5, 10, 20, 40, \ldots$ , you can divide each term by the previous term to find the common ratio:

$\frac{10}{5} = 2$
$\frac{20}{10} = 2$
$\frac{40}{20} = 2$

As you can see, the common ratio is $2$ .

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

A: To find the next term in a geometric sequence, you can multiply the last term by the common ratio. For example, if the sequence is $5, 10, 20, 40, \ldots$ and the common ratio is $2$ , you can multiply the last term by the common ratio to find the next term:

$40 \times 2 = 80$

Therefore, the next term in the sequence is $80$ .

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

A: The formula for finding the nth term of a geometric sequence is:

$a_n = a_1 \times r^{n-1}$

where $a_n$ is the nth term, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.

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:

$S_n = \frac{a_1(1-r^n)}{1-r}$

where $S_n$ is the sum of the first n terms, $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number.

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, on the other hand, is a type of sequence where each term after the first is found by adding a fixed number called the common difference.

Q: Can I use a geometric sequence to model real-world phenomena?

A: Yes, geometric sequences can be used to model real-world phenomena such as population growth, financial investments, and electrical circuits.

Q: What are some common applications of geometric sequences?

A: Geometric sequences have many real-world applications, including:

Finance: Geometric sequences are used to calculate interest rates and investment returns.
Economics: Geometric sequences are used to model economic growth and inflation.
Engineering: Geometric sequences are used to design and optimize systems, such as electrical circuits and mechanical systems.

Q: How do I determine if a sequence is geometric or not?

A: To determine if a sequence is geometric or not, you can check if each term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. If this is the case, then the sequence is geometric.

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

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

Not checking if the sequence is geometric before trying to find the next term or the sum.
Not using the correct formula for finding the nth term or the sum.
Not checking if the common ratio is valid (i.e., not equal to 1).

Conclusion

In conclusion, geometric sequences are a powerful tool for modeling real-world phenomena and solving problems in mathematics and other fields. By understanding the properties and applications of geometric sequences, you can develop a deeper appreciation for the beauty and utility of mathematics.