In the following number series, exactly one number is incorrect. Identify the wrong number:

21, 29, 45, 73, 125, 225, 421

Question

In the following number series, exactly one number is incorrect. Identify the wrong number:

21, 29, 45, 73, 125, 225, 421

Accepted Answer

**Solution:**

1. Let's analyze the pattern in the series: 21, 29, 45, 73, 125, 225, 421.

2. Observe the differences between consecutive terms:
   - 29 - 21 = 8
   - 45 - 29 = 16
   - 73 - 45 = 28
   - 125 - 73 = 52
   - 225 - 125 = 100
   - 421 - 225 = 196

3. The differences are: 8, 16, 28, 52, 100, 196. These do not form a simple arithmetic or geometric progression.

4. Let's check if the differences themselves follow a pattern. Notice that 8, 16, 28, 52, 100, 196 roughly double each time but not exactly.

5. Alternatively, consider the possibility that the series is based on powers of 2 minus 1 or some other formula. Let's check the terms against powers of 2:
   - 21 is suspicious because it does not fit well.

6. Let's try to find the pattern by considering the series as sums of powers of 2:
   - 21 = 2^4 + 2^2 + 1 = 16 + 4 + 1 = 21
   - 29 = 2^4 + 2^3 + 2^2 + 1 = 16 + 8 + 4 + 1 = 29
   - 45 = 32 + 8 + 4 + 1 = 45 (2^5 + 2^3 + 2^2 + 1)
   - 73 = 64 + 8 + 1 = 73 (2^6 + 2^3 + 1)
   - 125 = 64 + 32 + 16 + 8 + 4 + 1 = 125 (2^6 + 2^5 + 2^4 + 2^3 + 2^2 + 1)
   - 225 = 128 + 64 + 32 + 1 = 225 (2^7 + 2^6 + 2^5 + 1)
   - 421 is not a sum of powers of 2 plus 1 in a simple way.

7. Since this approach is complicated, let's check the ratio of consecutive terms:
   - 29/21 ≈ 1.38
   - 45/29 ≈ 1.55
   - 73/45 ≈ 1.62
   - 125/73 ≈ 1.71
   - 225/125 = 1.8
   - 421/225 ≈ 1.87

8. The ratio is increasing approximately by 0.1 each time, which suggests the first term 21 is too low to fit this pattern.

9. If we replace 21 with 27, the ratios become closer to a smooth progression:
   - 29/27 ≈ 1.07
   - 45/29 ≈ 1.55 (still irregular)

10. Given the options and the irregularities, the first term 21 is the odd one out and does not fit the pattern.

**Answer:** 21

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