To solve the problem of identifying the wrong number in the series: 7, 12, 40, 222, 1742, 17390, 208608, we will analyze the pattern in the series step by step.
### Step 1: Identify the pattern
We start by examining how each number relates to the next one in the series. The numbers are increasing significantly, suggesting that multiplication and subtraction are involved.
### Step 2: Analyze the first few terms
1. **From 7 to 12**:
- If we multiply 7 by 2, we get 14.
- To get 12, we subtract 2 from 14:
- \(7 \times 2 - 2 = 12\)
2. **From 12 to 40**:
- If we multiply 12 by 4, we get 48.
- To get 40, we subtract 8 from 48:
- \(12 \times 4 - 8 = 40\)
### Step 3: Continue the pattern
3. **From 40 to 222**:
- If we multiply 40 by 6, we get 240.
- To get 222, we subtract 18 from 240:
- \(40 \times 6 - 18 = 222\)
4. **From 222 to 1742**:
- If we multiply 222 by 8, we get 1776.
- To get 1742, we subtract 34 from 1776:
- \(222 \times 8 - 34 = 1742\)
### Step 4: Check the next term
5. **From 1742 to 17390**:
- If we multiply 1742 by 10, we get 17420.
- To get 17390, we subtract 30 from 17420:
- \(1742 \times 10 - 30 = 17390\)
### Step 5: Identify the inconsistency
Now, we notice that the pattern of multiplication and subtraction is consistent until we reach 1742. If we follow the pattern of multiplying by increasing even numbers (2, 4, 6, 8, 10, ...), we should have:
- The subtraction sequence appears to be increasing as well: -2, -8, -18, -34, ...
However, the next expected subtraction should follow the pattern:
- The differences in the subtractions are:
- From -2 to -8: increase of 6
- From -8 to -18: increase of 10
- From -18 to -34: increase of 16
Following this pattern, the next subtraction should be:
- -34 + 18 = -52
Thus, the calculation for the next term should be:
- \(1742 \times 10 - 52 = 17420 - 52 = 17368\)
### Conclusion
The number 1742 is incorrect in the series. The correct number should be 1744. Therefore, the wrong number in the series is **1742**.
