Understanding the Number Series Problem

The question asks us to find the missing number in the given series: 0, 2, 6, 12, ?, 30, 42. To solve this type of number series problem, we need to identify the underlying pattern or rule that generates the sequence of numbers.

Analyzing the Number Series Pattern

Let's look at the differences between consecutive terms in the series. This is a common method to find patterns in number series.

• Difference between the 2nd and 1st term: $2 - 0 = 2$
• Difference between the 3rd and 2nd term: $6 - 2 = 4$
• Difference between the 4th and 3rd term: $12 - 6 = 6$
• Difference between the 6th and 5th term: $30 - ?$
• Difference between the 7th and 6th term: $42 - 30 = 12$

The differences we have calculated so far are 2, 4, and 6. The last difference is 12. Let's look at the sequence of differences:

2, 4, 6, ?, ?, 12

This sequence of differences appears to be consecutive even numbers. If this pattern continues, the missing differences should be 8 and 10.

Sequence of differences: 2, 4, 6, 8, 10, 12

Calculating the Missing Number

Based on the pattern of differences, the difference between the 5th term (the missing number) and the 4th term (12) should be the next even number in the sequence, which is 8.

Missing Term = 4th Term + Difference

Missing Term = $12 + 8$

Missing Term = $20$

Let's verify this with the next term. If the 5th term is 20, the difference between the 6th term (30) and the 5th term (20) should be the next even number after 8, which is 10.

$30 - 20 = 10$

This confirms our pattern of adding consecutive even numbers (2, 4, 6, 8, 10, 12) to get the next term in the series.

Alternative Pattern: Product of Consecutive Numbers

Another way to look at this series is by expressing each term as a product of consecutive integers.

• Term 1: $0 = 0 \times 1$
• Term 2: $2 = 1 \times 2$
• Term 3: $6 = 2 \times 3$
• Term 4: $12 = 3 \times 4$
• Term 5: ?
• Term 6: $30 = 5 \times 6$
• Term 7: $42 = 6 \times 7$

If we index the terms starting from $n=0$ for the first number, the pattern for the $n$-th term seems to be $n \times (n+1)$.

• $n=0$: $0 \times (0+1) = 0 \times 1 = 0$
• $n=1$: $1 \times (1+1) = 1 \times 2 = 2$
• $n=2$: $2 \times (2+1) = 2 \times 3 = 6$
• $n=3$: $3 \times (3+1) = 3 \times 4 = 12$
• $n=4$: $4 \times (4+1) = 4 \times 5 = 20$
• $n=5$: $5 \times (5+1) = 5 \times 6 = 30$
• $n=6$: $6 \times (6+1) = 6 \times 7 = 42$

According to this pattern, the 5th term (which corresponds to $n=4$ in this indexing) is $4 \times 5 = 20$. Both patterns lead to the same answer.

Conclusion

Based on the analysis of the differences between terms (adding consecutive even numbers) and the product of consecutive numbers pattern, the missing number in the series 0, 2, 6, 12, ?, 30, 42 is 20.

Revision Table: Number Series Patterns

Additional Information: Solving Number Series Questions

Solving number series questions requires observation and recognizing common patterns. Here are some tips:

• Always check the differences between consecutive terms. If the first differences don't show a clear pattern, check the differences of the differences (double difference).
• Look for patterns involving squares, cubes, or roots of numbers.
• Consider patterns involving multiplication or division.
• Sometimes terms are related to the sum or product of previous terms (like Fibonacci series).
• Test multiple possible patterns if one doesn't immediately fit the entire series.
• Practice different types of series to become familiar with common patterns.

Understanding these techniques helps in quickly identifying the rule governing the number series and finding the missing term.