In the following number series only one is wrong. Find out the wrong number.
1, 5, 13, 31, 61, 125, 253

Options

A. 1
B. 5
C. 31
D. 61

Answer

To find the wrong number in the series 1, 5, 13, 31, 61, 125, 253, we will analyze the pattern in the series step by step.

### Step 1: Identify the pattern
Let's look at the numbers in the series:
1, 5, 13, 31, 61, 125, 253.

We will check if there is a consistent pattern in how the numbers are generated.

### Step 2: Check the differences between consecutive terms
Calculate the differences between consecutive terms:
- 5 - 1 = 4
- 13 - 5 = 8
- 31 - 13 = 18
- 61 - 31 = 30
- 125 - 61 = 64
- 253 - 125 = 128

So, the differences are:
4, 8, 18, 30, 64, 128.

### Step 3: Analyze the differences
Now, let's analyze the differences:
- The first difference is 4.
- The second difference is 8.
- The third difference is 18.
- The fourth difference is 30.
- The fifth difference is 64.
- The sixth difference is 128.

### Step 4: Look for a pattern in the differences
The differences themselves do not seem to follow a simple arithmetic progression, so let's look for a pattern in how we can generate the next number.

### Step 5: Check the multiplication and addition pattern
Let's try to see if there's a multiplication and addition pattern:
- Start with 1:
- 1 * 2 + 3 = 5
- Next:
- 5 * 2 + 3 = 13
- Next:
- 13 * 2 + 5 = 31
- Next:
- 31 * 2 + 5 = 61
- Next:
- 61 * 2 + 3 = 125
- Next:
- 125 * 2 + 3 = 253

### Step 6: Identify the inconsistency
From the calculations:
- The transition from 13 to 31 follows the pattern (13 * 2 + 5 = 31).
- However, the transition from 31 to 61 should follow the same pattern, which is (31 * 2 + 3 = 65), but we have 61 instead.

### Conclusion
Thus, the wrong number in the series is **31**.