Find the missing number in the series:

$$32, 40, 53, 76, ?, 172$$

Question

Find the missing number in the series:

$$32, 40, 53, 76, ?, 172$$

Accepted Answer

**Solution:**

1. Observe the differences between consecutive terms:
   - $40 - 32 = 8$
   - $53 - 40 = 13$
   - $76 - 53 = 23$

2. The differences are $8, 13, 23$. Let's check if these differences follow a pattern.

3. Check the differences between these differences:
   - $13 - 8 = 5$
   - $23 - 13 = 10$

4. The second-level differences are $5$ and $10$, which seem to be increasing by $5$.

5. Assuming the next second-level difference is $15$, the next first-level difference will be:
   $$23 + 15 = 38$$

6. Therefore, the missing term is:
   $$76 + 38 = 114$$

7. To verify, check the difference between $114$ and $172$:
   $$172 - 114 = 58$$

8. The next second-level difference would be $20$ (increasing by $5$ again), so the difference after $38$ should be:
   $$38 + 20 = 58$$

This confirms the pattern.

**Answer:** $114$

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