12 34 54 71 84 ?

Options

A. 89
B. 92
C. 124
D. 106

Answer

To solve the given series: 12, 34, 54, 71, 84, ?, we will analyze the differences between consecutive numbers to identify a pattern.

### Step 1: Find the differences between consecutive terms
- The difference between 34 and 12 is:
\( 34 - 12 = 22 \)

- The difference between 54 and 34 is:
\( 54 - 34 = 20 \)

- The difference between 71 and 54 is:
\( 71 - 54 = 17 \)

- The difference between 84 and 71 is:
\( 84 - 71 = 13 \)

So, the differences we have so far are:
- 22, 20, 17, 13

### Step 2: Find the differences of the differences
Now, let's find the differences of these differences:
- The difference between 20 and 22 is:
\( 20 - 22 = -2 \)

- The difference between 17 and 20 is:
\( 17 - 20 = -3 \)

- The difference between 13 and 17 is:
\( 13 - 17 = -4 \)

So, the second differences are:
- -2, -3, -4

### Step 3: Identify the pattern in the second differences
The second differences are decreasing by 1 each time:
- From -2 to -3 is a decrease of 1
- From -3 to -4 is also a decrease of 1

Following this pattern, the next difference should be:
- -5

### Step 4: Calculate the next difference
Now, we can find the next difference in the first series:
- The last difference we calculated was 13. If we subtract 5 from 13, we get:
\( 13 - 5 = 8 \)

### Step 5: Add the new difference to the last term
Now, we will add this new difference (8) to the last term in the original series (84):
- \( 84 + 8 = 92 \)

### Conclusion
Thus, the value of the question mark is **92**.