The pattern in the stages shows that the number of circles in each stage doubles the previous one. This reflects the formula:
- Stage 1: \(2^1 = 2\)
- Stage 2: \(2^2 = 4\)
- Stage 3: \(2^3 = 8\)
- Stage 4: \(2^4 = 16\)
Following this pattern, we can generalize that the number of circles in Stage \(n\) is given by \(2^n\).
Now, let's calculate the number of circles for stages 10, 15, 20, and 25:
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Stage 10: \[ 2^{10} = 1024 \]
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Stage 15: \[ 2^{15} = 32768 \]
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Stage 20: \[ 2^{20} = 1048576 \]
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Stage 25: \[ 2^{25} = 33554432 \]
Now let's find out which stage is closest to 30,000,000 circles:
- Stage 10: 1024
- Stage 15: 32768
- Stage 20: 1,048,576
- Stage 25: 33,554,432
Next, we can see that the number of circles in Stage 20 (1,048,576) is far less than 30,000,000, and the number in Stage 25 (33,554,432) exceeds it.
To find the closest:
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Distance from Stage 20 (1,048,576) to 30,000,000: \[ 30,000,000 - 1,048,576 = 28,951,424 \]
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Distance from Stage 25 (33,554,432) to 30,000,000: \[ 33,554,432 - 30,000,000 = 3,554,432 \]
Thus, Stage 25 is much closer to 30,000,000 than Stage 20.
Therefore, the stage that has circles closest to 30,000,000 is Stage 25.
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