To find the wrong number in the series: 8, 13, 21, 32, 47, 63, 83, we will analyze the differences between consecutive terms.
### Step-by-Step Solution:
1. **Identify the Series**:
The given series is:
\[
8, 13, 21, 32, 47, 63, 83
\]
2. **Calculate the Differences**:
We will find the differences between consecutive terms:
- \( 13 - 8 = 5 \)
- \( 21 - 13 = 8 \)
- \( 32 - 21 = 11 \)
- \( 47 - 32 = 15 \)
- \( 63 - 47 = 16 \)
- \( 83 - 63 = 20 \)
So the differences are:
\[
5, 8, 11, 15, 16, 20
\]
3. **Analyze the Differences**:
Now, we will analyze the differences:
- The first difference is \( 5 \)
- The second difference is \( 8 \) (which is \( 5 + 3 \))
- The third difference is \( 11 \) (which is \( 8 + 3 \))
- The fourth difference is \( 15 \) (which is \( 11 + 4 \))
- The fifth difference is \( 16 \) (which is \( 15 + 1 \))
- The sixth difference is \( 20 \) (which is \( 16 + 4 \))
We notice that the differences do not follow a consistent pattern after the third term.
4. **Identify the Anomaly**:
The differences seem to increase by an inconsistent amount. If we consider the pattern of increasing by 3, we would expect:
- After \( 11 \), the next difference should be \( 14 \) (not \( 15 \)).
- If we replace \( 47 \) with \( 46 \), the series would be \( 8, 13, 21, 32, 46, 63, 83 \).
5. **Verify the Corrected Series**:
Let's check the new differences:
- \( 13 - 8 = 5 \)
- \( 21 - 13 = 8 \)
- \( 32 - 21 = 11 \)
- \( 46 - 32 = 14 \)
- \( 63 - 46 = 17 \)
- \( 83 - 63 = 20 \)
The new differences are:
\[
5, 8, 11, 14, 17, 20
\]
This shows a consistent increase of 3.
6. **Conclusion**:
Therefore, the wrong number in the series is \( 47 \). The correct number should be \( 46 \).
### Final Answer:
The wrong number in the series is **47**.
