Find the wrong number in each of the following number series.
3 5 10 20 39 63 100.

Options

A. 63
B. 5
C. 20
D. 39

Answer

To find the wrong number in the series 3, 5, 10, 20, 39, 63, 100, we can analyze the pattern in the series step by step.

### Step-by-Step Solution:

1. **Identify the Series**: The given series is:
\[ 3, 5, 10, 20, 39, 63, 100 \]

2. **Find the Differences**: Calculate the differences between consecutive terms:
- \(5 - 3 = 2\)
- \(10 - 5 = 5\)
- \(20 - 10 = 10\)
- \(39 - 20 = 19\)
- \(63 - 39 = 24\)
- \(100 - 63 = 37\)

So, the differences are:
\[ 2, 5, 10, 19, 24, 37 \]

3. **Analyze the Differences**: Now, let's look for a pattern in the differences:
- The first difference is \(2\) (which is \(1^2 + 1\))
- The second difference is \(5\) (which is \(2^2 + 1\))
- The third difference is \(10\) (which is \(3^2 + 1\))
- The fourth difference is \(19\) (which is \(4^2 + 3\))
- The fifth difference is \(24\) (which is \(5^2 - 1\))
- The sixth difference is \(37\) (which is \(6^2 + 1\))

4. **Identify the Pattern**: The expected differences based on the pattern of \(n^2 + 1\) should be:
- For \(n=1\): \(1^2 + 1 = 2\)
- For \(n=2\): \(2^2 + 1 = 5\)
- For \(n=3\): \(3^2 + 1 = 10\)
- For \(n=4\): \(4^2 + 1 = 17\) (but we have \(19\))
- For \(n=5\): \(5^2 + 1 = 26\) (but we have \(24\))
- For \(n=6\): \(6^2 + 1 = 37\)

5. **Find the Wrong Number**: The difference for \(n=4\) should be \(17\) instead of \(19\). Therefore, the term \(39\) is incorrect because:
\[ 20 + 17 = 37 \]
Instead of \(39\).

6. **Conclusion**: The wrong number in the series is:
\[ \text{Wrong Number: } 39 \]