To solve the problem of filling in the blank in the sequence 0, 2, 6, __, 20, 30, 42, we can follow these steps:
### Step 1: Identify the given sequence
The sequence provided is:
0, 2, 6, __, 20, 30, 42.
### Step 2: Calculate the differences between consecutive terms
Let's find the differences between the numbers in the sequence:
- Difference between 0 and 2: 2 - 0 = 2
- Difference between 2 and 6: 6 - 2 = 4
- Difference between 6 and the blank: Let’s denote the blank as x, so the difference is x - 6.
- Difference between x and 20: 20 - x.
- Difference between 20 and 30: 30 - 20 = 10
- Difference between 30 and 42: 42 - 30 = 12
So the differences we have so far are:
2, 4, (x - 6), (20 - x), 10, 12.
### Step 3: Look for a pattern in the differences
The differences we have are:
- 2
- 4
- (x - 6)
- (20 - x)
- 10
- 12
Now, let's analyze the differences we have:
- The first difference is 2.
- The second difference is 4.
- The third difference (x - 6) should logically follow the pattern.
### Step 4: Establish a pattern for the differences
From the differences we see:
- The first two differences are increasing by 2 (2 to 4).
- If we continue this pattern, the next difference could be 6 (which would make x - 6 = 6).
- Therefore, we can set up the equation: x - 6 = 6.
### Step 5: Solve for x
Now, solve for x:
x - 6 = 6
=> x = 6 + 6
=> x = 12.
### Step 6: Verify the next difference
Now, we need to verify the next difference:
- The difference between 12 and 20 should be: 20 - 12 = 8.
- This fits the pattern as the next difference after 6 should be 8.
### Conclusion
Thus, the number that fills in the blank is **12**.
