To find the missing number in the series: 5, 7, 18, 47, 103, ?, we will analyze the differences between the consecutive terms.
### Step 1: Identify the series
The series is:
5, 7, 18, 47, 103, ?
### Step 2: Calculate the differences between consecutive terms
- Difference between 7 and 5:
\( 7 - 5 = 2 \)
- Difference between 18 and 7:
\( 18 - 7 = 11 \)
- Difference between 47 and 18:
\( 47 - 18 = 29 \)
- Difference between 103 and 47:
\( 103 - 47 = 56 \)
So, the differences are:
- 2, 11, 29, 56
### Step 3: Calculate the differences of the differences
Now, we will find the differences of these differences:
- Difference between 11 and 2:
\( 11 - 2 = 9 \)
- Difference between 29 and 11:
\( 29 - 11 = 18 \)
- Difference between 56 and 29:
\( 56 - 29 = 27 \)
So, the second level differences are:
- 9, 18, 27
### Step 4: Calculate the differences of the second level differences
Next, we will find the differences of these second level differences:
- Difference between 18 and 9:
\( 18 - 9 = 9 \)
- Difference between 27 and 18:
\( 27 - 18 = 9 \)
The third level differences are:
- 9, 9
### Step 5: Predict the next second level difference
Since the third level differences are constant (9), we can predict that the next second level difference will also be:
- \( 27 + 9 = 36 \)
### Step 6: Predict the next first level difference
Now, we can add this new second level difference to the last first level difference:
- \( 56 + 36 = 92 \)
### Step 7: Predict the next term in the series
Finally, we can add this first level difference to the last term in the original series:
- \( 103 + 92 = 195 \)
Thus, the missing number in the series is **195**.
### Summary
The missing number in the series 5, 7, 18, 47, 103, ? is **195**.
