Find the missing number in the series:

$$48, 55, 76, ?, 160, 223$$

Question

Find the missing number in the series:

$$48, 55, 76, ?, 160, 223$$

Accepted Answer

**Solution:**

1. Observe the series: $$48, 55, 76, ?, 160, 223$$

2. Calculate the differences between consecutive terms:
   - $55 - 48 = 7$
   - $76 - 55 = 21$

3. Check the difference between the last two known terms:
   - $223 - 160 = 63$

4. Notice the pattern in differences: $7, 21, ?, 63$
   - These differences seem to be multiplied by 3 each time: $7 	imes 3 = 21$, $21 	imes 3 = 63$

5. So, the missing difference should be $21 	imes 3 = 63$

6. Find the missing term by adding this difference to the previous term:
   - Missing term = $76 + 35 = 111$

7. Verify the next difference:
   - $160 - 111 = 49$ (which does not fit the previous pattern exactly, so let's re-examine)

**Re-examining the differences:**
- Differences: $7, 21, ?, 160 - ?$, $223 - 160 = 63$

Try another approach:
- Differences: $7, 21, x, y, 63$

Check if differences are multiples of 7:
- $7 = 7 	imes 1$
- $21 = 7 	imes 3$
- $63 = 7 	imes 9$

The multipliers are $1, 3, ?, ?, 9$ which suggests a pattern in multipliers: $1, 3, 5, 7, 9$

So,
- 3rd difference = $7 	imes 5 = 35$
- 4th difference = $7 	imes 7 = 49$

Calculate missing terms:
- 3rd term to 4th term: $76 + 35 = 111$
- 4th term to 5th term: $111 + 49 = 160$

This matches the series perfectly.

**Answer:** $111$

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