What will come in place of question mark (?) in the given number series?
14 16 22 34 54 ?

Options

A. 76
B. 84
C. 86
D. 74

Answer

To solve the problem of finding the missing number in the series: 14, 16, 22, 34, 54, ?, we will analyze the differences between the consecutive numbers in the series.

### Step 1: Identify the differences between consecutive terms
- The difference between 16 and 14 is:
\( 16 - 14 = 2 \)
- The difference between 22 and 16 is:
\( 22 - 16 = 6 \)
- The difference between 34 and 22 is:
\( 34 - 22 = 12 \)
- The difference between 54 and 34 is:
\( 54 - 34 = 20 \)

### Step 2: Write down the differences
Now we have the differences:
- 2, 6, 12, 20

### Step 3: Identify the pattern in the differences
Next, we will find the differences of these differences:
- The difference between 6 and 2 is:
\( 6 - 2 = 4 \)
- The difference between 12 and 6 is:
\( 12 - 6 = 6 \)
- The difference between 20 and 12 is:
\( 20 - 12 = 8 \)

### Step 4: Write down the second level of differences
Now we have the second level of differences:
- 4, 6, 8

### Step 5: Identify the pattern in the second level of differences
The differences of the second level are:
- The difference between 6 and 4 is:
\( 6 - 4 = 2 \)
- The difference between 8 and 6 is:
\( 8 - 6 = 2 \)

This indicates that the second level of differences is increasing by 2 each time.

### Step 6: Calculate the next difference in the first level
Following the pattern, the next difference in the first level should be:
- The last difference was 20, so we add the next increment (which is 10):
\( 20 + 10 = 30 \)

### Step 7: Calculate the missing term
Now, we can find the missing term (let's call it x) in the original series:
- The last known term is 54, so we add the difference we just calculated:
\( x = 54 + 30 = 84 \)

### Conclusion
Thus, the missing number in the series is **84**.

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