The number of circles at Stage 10 is extremely large. Write an expression to represent this number.

Question

1 2
2 4
3 8
4 16
5 32
10 1024

Answer 1

The pattern you provided shows that the number of circles doubles with each stage. Specifically, at Stage \( n \), the number of circles can be represented as \( 2^n \).

We can observe this from the examples given:

• Stage 1: \( 2^1 = 2 \)
• Stage 2: \( 2^2 = 4 \)
• Stage 3: \( 2^3 = 8 \)
• Stage 4: \( 2^4 = 16 \)
• Stage 5: \( 2^5 = 32 \)
• Stage 10: \( 2^{10} = 1024 \)

Therefore, the expression to represent the number of circles at Stage 10 is:

\[ 2^{10} \]

Calculating this gives us:

\[ 2^{10} = 1024 \]

Thus, the number of circles at Stage 10 is \( 1024 \).

Answer 2

How many times as many circles will there be at Stage 21 compared to Stage 20?

Answer 3

To find out how many times as many circles there will be at Stage 21 compared to Stage 20, we can use the expression for the number of circles at each stage, which is \( 2^n \) where \( n \) is the stage number.

1. At Stage 20, the number of circles is \( 2^{20} \).
2. At Stage 21, the number of circles is \( 2^{21} \).

Now, to determine how many times as many circles are at Stage 21 compared to Stage 20, we can set up the following ratio:

\[ \text{Ratio} = \frac{2^{21}}{2^{20}} \]

Using the properties of exponents, we simplify this ratio:

\[ \text{Ratio} = 2^{21 - 20} = 2^1 = 2 \]

Thus, there will be 2 times as many circles at Stage 21 compared to Stage 20.

Answer 4

Stage 23 Stage 23 will have more circles than Stage 20. Tyler believes that Stage 23 will have three times as many circles as Stage 20. Lin believes that Stage 23 will have eight times as many circles as Stage 20. Who is correct?

Answer 5

To determine who is correct regarding the number of circles at Stage 23 compared to Stage 20, we can calculate the number of circles at each stage using the formula \( 2^n \), where \( n \) is the stage number.

1. Number of circles at Stage 20: \[ 2^{20} \]

2. Number of circles at Stage 23: \[ 2^{23} \]

Now, we can find the ratio of the number of circles at Stage 23 to the number of circles at Stage 20:

\[ \text{Ratio} = \frac{2^{23}}{2^{20}} = 2^{23 - 20} = 2^3 = 8 \]

Conclusion:

• This means that Stage 23 has 8 times as many circles as Stage 20.
• Therefore, Lin is correct.

Answer 6

Here are Stages 1–4. What stage will have closest to 30,000,000 circles? Use the calculator at the top of this page if it helps you with your thinking.

Answer 7

The number of circles at each stage can be represented by the formula \( 2^n \), where \( n \) is the stage number. We want to find the stage \( n \) such that \( 2^n \) is closest to 30,000,000.

First, we'll calculate \( 2^n \) for various stages to see which stage gives us a number near 30,000,000:

• \( 2^{24} = 16,777,216 \)
• \( 2^{25} = 33,554,432 \)

Now, we see that:

• \( 2^{24} = 16,777,216 \) (which is less than 30,000,000)
• \( 2^{25} = 33,554,432 \) (which is greater than 30,000,000)

Next, let's determine how close each of these values is to 30,000,000:

• Difference for \( 2^{24} \): \[ 30,000,000 - 16,777,216 = 13,222,784 \]

• Difference for \( 2^{25} \): \[ 33,554,432 - 30,000,000 = 3,554,432 \]

Comparing the differences, \( 2^{25} \) is closer to 30,000,000 than \( 2^{24} \).

Conclusion:

The stage that will have circles closest to 30,000,000 is Stage 25.

Answer 8

2^17 is how many times as many as 2^13

Answer 9

To find how many times \( 2^{17} \) is compared to \( 2^{13} \), we can calculate the ratio:

\[ \text{Ratio} = \frac{2^{17}}{2^{13}} \]

Using the properties of exponents, we can simplify this:

\[ \text{Ratio} = 2^{17 - 13} = 2^4 \]

Now calculating \( 2^4 \):

\[ 2^4 = 16 \]

Therefore, \( 2^{17} \) is 16 times as many as \( 2^{13} \).