To find the missing number in the series: 0, 2, 6, ?, 20, 30, 42, we can analyze the pattern of the differences between the consecutive numbers.
**Step 1: Identify the differences between consecutive terms.**
- The difference between the first term (0) and the second term (2) is:
\( 2 - 0 = 2 \)
- The difference between the second term (2) and the third term (6) is:
\( 6 - 2 = 4 \)
- The difference between the third term (6) and the fourth term (?), which we need to find, is unknown.
- The difference between the fourth term (?) and the fifth term (20) is:
\( 20 - ? \)
- The difference between the fifth term (20) and the sixth term (30) is:
\( 30 - 20 = 10 \)
- The difference between the sixth term (30) and the seventh term (42) is:
\( 42 - 30 = 12 \)
**Step 2: Write down the differences we have calculated.**
So far, we have the following differences:
- From 0 to 2: \( +2 \)
- From 2 to 6: \( +4 \)
- From 6 to ?: \( ? \)
- From ? to 20: \( ? \)
- From 20 to 30: \( +10 \)
- From 30 to 42: \( +12 \)
**Step 3: Analyze the pattern of differences.**
The differences we have are 2, 4, ?, ?, 10, 12.
Let’s look at the differences we have:
- The first difference is 2.
- The second difference is 4.
- The next difference should logically follow the pattern of increasing by 2 each time.
If we continue this pattern:
- After 4, the next difference would be \( 4 + 2 = 6 \).
- After 6, the next difference would be \( 6 + 2 = 8 \).
So, the differences should be:
- 2, 4, 6, 8, 10, 12.
**Step 4: Find the missing number.**
Now we can fill in the missing differences:
- From 6 to ?: \( +6 \)
- From ? to 20: \( +8 \)
Now we can find the missing number:
- Starting from 6, if we add 6, we get:
\( 6 + 6 = 12 \)
Thus, the missing number in the series is **12**.
**Final Series:**
0, 2, 6, 12, 20, 30, 42.
**Step 5: Conclusion**
The number that fits in the question mark (?) place is **12**.
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