Identify the incorrect number in the series:

$$380, 381, 387, 413, 476, 600, 815$$

Question

Identify the incorrect number in the series:

$$380, 381, 387, 413, 476, 600, 815$$

Accepted Answer

**Solution:**

1. Observe the differences between consecutive terms:
   - $381 - 380 = 1$
   - $387 - 381 = 6$
   - $413 - 387 = 26$
   - $476 - 413 = 63$
   - $600 - 476 = 124$
   - $815 - 600 = 215$

2. The differences do not follow a simple arithmetic or geometric pattern, so let's check if the differences themselves form a recognizable pattern.

3. Consider the differences: $1, 6, 26, 63, 124, 215$

4. Check if these differences correspond to any pattern such as cubes or squares:
   - $1 = 1^3$
   - $6 = 2^3 - 2$
   - $26$ is close to $3^3 = 27$
   - $63 = 4^3 - 1$
   - $124$ is close to $5^3 = 125$
   - $215 = 6^3 - 1$

5. The differences seem irregular, so try checking the ratio of consecutive differences:
   - $6/1 = 6$
   - $26/6 
eq$ integer
   - $63/26 
eq$ integer

6. Since the pattern is unclear, check the series for a simpler pattern by examining the numbers themselves.

7. Another approach: Check if the numbers are close to perfect cubes:
   - $7^3 = 343$, close to $380$
   - $7^3 + 38 = 381$
   - $7^3 + 44 = 387$

8. This approach is complicated; instead, check if the number $381$ fits the increasing pattern.

9. The jump from $380$ to $381$ is $+1$, but the next jump is $+6$, then $+26$, which is inconsistent.

10. If we replace $381$ with $386$, the differences become:
    - $386 - 380 = 6$
    - $387 - 386 = 1$

This seems more consistent with the pattern of small and large jumps alternating.

11. Therefore, $381$ is the incorrect number in the series.

**Answer:** $381$

Answer

Answer

Answer

Answer