13, 15, 25, 55, 123, ?

Options

A. 190
B. 234
C. 293
D. 253

Explanation

To find the missing number in the series: 13, 15, 25, 55, 123, ?, we will analyze the pattern in the sequence step by step.

### Step 1: Identify the differences between consecutive terms
- The difference between 15 and 13 is:
\( 15 - 13 = 2 \)

- The difference between 25 and 15 is:
\( 25 - 15 = 10 \)

- The difference between 55 and 25 is:
\( 55 - 25 = 30 \)

- The difference between 123 and 55 is:
\( 123 - 55 = 68 \)

### Step 2: List the differences
Now we have the differences:
- From 13 to 15: 2
- From 15 to 25: 10
- From 25 to 55: 30
- From 55 to 123: 68

So, the differences are: 2, 10, 30, 68.

### Step 3: Identify the pattern in the differences
Next, we will analyze the differences: 2, 10, 30, 68.
- The difference between 10 and 2 is:
\( 10 - 2 = 8 \)

- The difference between 30 and 10 is:
\( 30 - 10 = 20 \)

- The difference between 68 and 30 is:
\( 68 - 30 = 38 \)

### Step 4: List the second differences
Now we have the second differences:
- From 2 to 10: 8
- From 10 to 30: 20
- From 30 to 68: 38

So, the second differences are: 8, 20, 38.

### Step 5: Identify the pattern in the second differences
Next, we will analyze the second differences: 8, 20, 38.
- The difference between 20 and 8 is:
\( 20 - 8 = 12 \)

- The difference between 38 and 20 is:
\( 38 - 20 = 18 \)

### Step 6: List the third differences
Now we have the third differences:
- From 8 to 20: 12
- From 20 to 38: 18

So, the third differences are: 12, 18.

### Step 7: Identify the pattern in the third differences
The difference between 18 and 12 is:
\( 18 - 12 = 6 \)

This indicates that the third differences are increasing by 6.

### Step 8: Predict the next third difference
If we continue this pattern, the next third difference would be:
\( 18 + 6 = 24 \)

### Step 9: Predict the next second difference
Adding this to the last second difference:
\( 38 + 24 = 62 \)

### Step 10: Predict the next first difference
Now, we add this to the last first difference:
\( 68 + 62 = 130 \)

### Step 11: Predict the next term in the original series
Finally, we add this to the last term in the series:
\( 123 + 130 = 253 \)

Thus, the missing number in the series is **253**.

### Summary of Steps
1. Calculate the first differences between terms.
2. Calculate the second differences.
3. Calculate the third differences.
4. Identify the pattern in the differences.
5. Predict the next term based on the identified pattern.