Write an Equation to Describe the Sequence Below and Then Find the 5th Term

Oct 13, 2025 by magicalceylontours.com 50 views

Write an Equation to Describe the Sequence Below and Then Find the 5th Term

Understanding the Sequence

The given sequence is 10, 20, 40, and we are asked to write an equation to describe this sequence and then find the 5th term. To start, let's analyze the pattern of the sequence. We can see that each term is obtained by multiplying the previous term by 2.

Identifying the Pattern

Let's examine the differences between consecutive terms:

20 - 10 = 10
40 - 20 = 20

We can observe that the differences between consecutive terms are doubling. This suggests that the sequence is obtained by multiplying the previous term by 2.

Writing the Equation

Based on the pattern we identified, we can write the equation for the nth term as:

an = 2(n - 1)

This equation states that the nth term is equal to 2 times the position of the term minus 1.

Verifying the Equation

Let's verify the equation by plugging in the values of n for the given terms:

For n = 1, an = 2(1 - 1) = 2(0) = 0 (not equal to 10, so we need to adjust the equation)
For n = 2, an = 2(2 - 1) = 2(1) = 2 (not equal to 20, so we need to adjust the equation)
For n = 3, an = 2(3 - 1) = 2(2) = 4 (not equal to 40, so we need to adjust the equation)

It seems that the equation an = 2(n - 1) is not correct. Let's try to adjust it.

Adjusting the Equation

Let's re-examine the sequence and try to find a pattern:

10 = 2^3 - 2
20 = 2^4 - 2
40 = 2^5 - 2

We can see that each term is equal to 2 raised to the power of the position minus 2.

New Equation

Based on the pattern we identified, we can write the new equation for the nth term as:

an = 2^n - 2

This equation states that the nth term is equal to 2 raised to the power of the position minus 2.

Verifying the New Equation

Let's verify the new equation by plugging in the values of n for the given terms:

For n = 1, an = 2^1 - 2 = 0 (not equal to 10, so we need to adjust the equation)
For n = 2, an = 2^2 - 2 = 2 (not equal to 20, so we need to adjust the equation)
For n = 3, an = 2^3 - 2 = 6 (not equal to 40, so we need to adjust the equation)

It seems that the new equation an = 2^n - 2 is not correct. Let's try to adjust it again.

Adjusting the New Equation

Let's re-examine the sequence and try to find a pattern:

10 = 2^3 - 2 + 2
20 = 2^4 - 2 + 2
40 = 2^5 - 2 + 2

We can see that each term is equal to 2 raised to the power of the position minus 2 plus 2.

Final Equation

Based on the pattern we identified, we can write the final equation for the nth term as:

an = 2^n - 2 + 2

This equation states that the nth term is equal to 2 raised to the power of the position minus 2 plus 2.

Simplifying the Final Equation

We can simplify the final equation by combining the constants:

an = 2^n

This equation states that the nth term is equal to 2 raised to the power of the position.

Finding the 5th Term

Now that we have the final equation, we can find the 5th term by plugging in n = 5:

a5 = 2^5 = 32

Therefore, the 5th term of the sequence is 32.

Conclusion

In this article, we wrote an equation to describe the given sequence and then found the 5th term. We started by analyzing the pattern of the sequence and identifying the differences between consecutive terms. We then wrote an equation for the nth term and verified it by plugging in the values of n for the given terms. After adjusting the equation several times, we finally arrived at the final equation an = 2^n, which states that the nth term is equal to 2 raised to the power of the position. We then used this equation to find the 5th term, which is 32.
Q&A: Understanding the Sequence and Finding the 5th Term

Q: What is the pattern of the sequence?

A: The pattern of the sequence is obtained by multiplying the previous term by 2.

Q: How do you write an equation to describe the sequence?

A: To write an equation to describe the sequence, we need to identify the pattern and then use it to create a formula for the nth term.

Q: What is the equation for the nth term?

A: The equation for the nth term is an = 2^n.

Q: How do you verify the equation?

A: To verify the equation, we need to plug in the values of n for the given terms and check if the equation holds true.

Q: What if the equation is not correct?

A: If the equation is not correct, we need to re-examine the sequence and try to find a new pattern. We can then use this new pattern to create a new equation for the nth term.

Q: How do you find the 5th term of the sequence?

A: To find the 5th term of the sequence, we need to plug in n = 5 into the equation an = 2^n.

Q: What is the 5th term of the sequence?

A: The 5th term of the sequence is 32.

Q: Can you explain the concept of exponential growth in the sequence?

A: Yes, the sequence exhibits exponential growth because each term is obtained by multiplying the previous term by 2. This means that the sequence grows rapidly as the position of the term increases.

Q: How does the equation an = 2^n relate to exponential growth?

A: The equation an = 2^n represents the exponential growth of the sequence because it shows that each term is equal to 2 raised to the power of the position.

Q: What are some real-world applications of exponential growth?

A: Exponential growth has many real-world applications, such as population growth, financial growth, and technological advancements.

Q: Can you provide an example of exponential growth in a real-world scenario?

A: Yes, a classic example of exponential growth is the growth of a population of bacteria. If a population of bacteria doubles every hour, the number of bacteria will grow exponentially over time.

Q: How can you use the equation an = 2^n to model real-world scenarios?

A: You can use the equation an = 2^n to model real-world scenarios that exhibit exponential growth, such as population growth, financial growth, and technological advancements.

Q: What are some limitations of the equation an = 2^n?

A: The equation an = 2^n is a simple model that assumes exponential growth without any limitations. In reality, growth is often limited by factors such as resources, competition, and environmental constraints.

Q: Can you provide a more realistic model of exponential growth?

A: Yes, a more realistic model of exponential growth would take into account the limitations of growth, such as resource constraints and competition. This could be represented by a logistic growth model, which is a more complex equation that takes into account the limitations of growth.