Find the missing number in the series:

$$80, \, ?, \, 117, \, 168, \, 271, \, 478$$

Question

Find the missing number in the series:

$$80, \, ?, \, 117, \, 168, \, 271, \, 478$$

Accepted Answer

**Solution:**

1. Observe the series: $$80, \, ?, \, 117, \, 168, \, 271, \, 478$$

2. Calculate the differences between consecutive known terms:
   - Between 117 and 168: $$168 - 117 = 51$$
   - Between 168 and 271: $$271 - 168 = 103$$
   - Between 271 and 478: $$478 - 271 = 207$$

3. Notice the pattern in the differences:
   - $$51, 103, 207$$
   - Each difference approximately doubles the previous one (51 × 2 = 102, close to 103; 103 × 2 = 206, close to 207)

4. Assuming the same pattern applies backward:
   - The difference before 51 should be about half of 51, i.e., around 25 or 26.

5. Calculate the missing number:
   - Let the missing number be *x*.
   - Difference between *x* and 80 should be approximately 12 or 13 (since 25 is double of 12.5).

6. Check the difference between 80 and *x*:
   - If difference is 12, then *x* = 80 + 12 = 92

7. Verify the pattern with differences:
   - 92 to 117: $$117 - 92 = 25$$
   - 117 to 168: 51 (given)
   - 168 to 271: 103
   - 271 to 478: 207

8. The differences form the sequence: 12, 25, 51, 103, 207, which roughly doubles each time.

**Answer:** $92$

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