Find the wrong number in the following series:

$$8, \, 21, \, 41, \, 72, \, 116, \, 175, \, 257$$

Question

Find the wrong number in the following series:

$$8, \, 21, \, 41, \, 72, \, 116, \, 175, \, 257$$

Accepted Answer

**Solution:**

1. Observe the series: $$8, 21, 41, 72, 116, 175, 257$$

2. Calculate the differences between consecutive terms:
   - $21 - 8 = 13$
   - $41 - 21 = 20$
   - $72 - 41 = 31$
   - $116 - 72 = 44$
   - $175 - 116 = 59$
   - $257 - 175 = 82$

3. Now, look at the differences: $$13, 20, 31, 44, 59, 82$$

4. Check the pattern in these differences:
   - $20 - 13 = 7$
   - $31 - 20 = 11$
   - $44 - 31 = 13$
   - $59 - 44 = 15$
   - $82 - 59 = 23$

The increments are irregular (7, 11, 13, 15, 23), which suggests an error.

5. If we replace $175$ with $177$, the differences become:
   - $177 - 116 = 61$
   - $257 - 177 = 80$

6. Now, the differences are:
   $$13, 20, 31, 44, 61, 80$$

7. The increments between these differences are:
   - $20 - 13 = 7$
   - $31 - 20 = 11$
   - $44 - 31 = 13$
   - $61 - 44 = 17$
   - $80 - 61 = 19$

These increments follow a pattern of increasing prime numbers: 7, 11, 13, 17, 19.

8. Therefore, the wrong number in the original series is $175$.

**Answer:** $175$

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