Find the missing term in the series:

25, ?, 256, 476, 745, 1050

Question

Find the missing term in the series:

25, ?, 256, 476, 745, 1050

Accepted Answer

**Solution:**

1. Observe the series: 25, ?, 256, 476, 745, 1050

2. Calculate the differences between the known consecutive terms starting from the third term:
   - Difference between 476 and 256 = 476 - 256 = 220
   - Difference between 745 and 476 = 745 - 476 = 269
   - Difference between 1050 and 745 = 1050 - 745 = 305

3. The differences are increasing but not by a fixed amount. Let's check the differences between these differences:
   - 269 - 220 = 49
   - 305 - 269 = 36

4. The second-level differences are 49 and 36, which are perfect squares (7² and 6²), but decreasing.

5. Let's check the difference between the missing term and 25:
   - Assume the missing term is *x*.
   - Difference between *x* and 25 = *x* - 25
   - Difference between 256 and *x* = 256 - *x*

6. The pattern of differences seems irregular, so let's try another approach by checking the differences between consecutive terms:

- Let the differences be:
     d1 = x - 25
     d2 = 256 - x
     d3 = 476 - 256 = 220
     d4 = 745 - 476 = 269
     d5 = 1050 - 745 = 305

7. Now, check if the differences follow a pattern:
   - d3 = 220
   - d4 = 269 (difference from d3 = 49)
   - d5 = 305 (difference from d4 = 36)

8. The differences between differences are 49 and 36, which are 7² and 6² respectively, decreasing by 13.

9. If this pattern continues backward:
   - Difference between d2 and d3 should be 64 (8²)
   - So, d2 = d3 - 64 = 220 - 64 = 156

- Difference between d1 and d2 should be 81 (9²)
   - So, d1 = d2 - 81 = 156 - 81 = 75

10. Now, calculate the missing term *x*:
    - d1 = x - 25 = 75  =>  x = 25 + 75 = 100

**Answer:** 100

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