Question

stage 1: 2 circles, stage 2: 4 circles, stage 3: 8 circles, stage 4: 16 circles. which stage will have the closest to 30,000,000 circles?

Answer 1

To determine which stage has the closest to 30,000,000 circles, we can observe the pattern:

- Stage 1: \(2^1 = 2\) circles
- Stage 2: \(2^2 = 4\) circles
- Stage 3: \(2^3 = 8\) circles
- Stage 4: \(2^4 = 16\) circles
- Stage 5: \(2^5 = 32\)
- Stage 6: \(2^6 = 64\)
- Stage 7: \(2^7 = 128\)
- Stage 8: \(2^8 = 256\)
- Stage 9: \(2^9 = 512\)
- Stage 10: \(2^{10} = 1024\)
- Stage 11: \(2^{11} = 2048\)
- Stage 12: \(2^{12} = 4096\)
- Stage 13: \(2^{13} = 8192\)
- Stage 14: \(2^{14} = 16384\)
- Stage 15: \(2^{15} = 32768\)
- Stage 16: \(2^{16} = 65536\)
- Stage 17: \(2^{17} = 131072\)
- Stage 18: \(2^{18} = 262144\)
- Stage 19: \(2^{19} = 524288\)
- Stage 20: \(2^{20} = 1048576\)
- Stage 21: \(2^{21} = 2097152\)
- Stage 22: \(2^{22} = 4194304\)
- Stage 23: \(2^{23} = 8388608\)
- Stage 24: \(2^{24} = 16777216\)
- Stage 25: \(2^{25} = 33554432\)

Now, we see that \(2^{24} = 16,777,216\) and \(2^{25} = 33,554,432\).

Since 30,000,000 is between these two values, we can calculate the distances:

- \(30,000,000 - 16,777,216 = 13,222,784\)
- \(33,554,432 - 30,000,000 = 3,554,432\)

3,554,432 is significantly smaller than 13,222,784, meaning that Stage 25 (which has 33,554,432 circles) is closer to 30,000,000 than Stage 24 (which has 16,777,216 circles).

Thus, the stage that will have the closest number of circles to 30,000,000 is **Stage 25**.