Understanding the Number Series Pattern

The question asks us to find the next number in the given series: 4, 22, 47, 83, 132, 198, 283, ?.

To solve number series problems, we often look for a pattern in the differences between consecutive terms. Let's calculate the differences between adjacent numbers in the series.

Calculating First Differences in the Series

We find the difference between each term and the one preceding it:

• $22 - 4 = 18$
• $47 - 22 = 25$
• $83 - 47 = 36$
• $132 - 83 = 49$
• $198 - 132 = 66$
• $283 - 198 = 85$

The first differences are: 18, 25, 36, 49, 66, 85.

Analyzing Second Differences

The first differences themselves form a new series. Let's find the differences between consecutive terms of this new series (the second differences):

• $25 - 18 = 7$
• $36 - 25 = 11$
• $49 - 36 = 13$
• $66 - 49 = 17$
• $85 - 66 = 19$

The second differences are: 7, 11, 13, 17, 19.

Identifying Pattern in Third Differences

Let's look at the differences between consecutive terms of the second differences (the third differences):

• $11 - 7 = 4$
• $13 - 11 = 2$
• $17 - 13 = 4$
• $19 - 17 = 2$

The third differences form a pattern: 4, 2, 4, 2. This is an alternating sequence of 4 and 2.

Predicting the Next Terms

Following the pattern of the third differences (4, 2, 4, 2, ...), the next third difference should be 4.

Using this, we can find the next second difference:

• Next second difference = Last second difference + Next third difference
• Next second difference = $19 + 4 = 23$

Now, we can find the next first difference:

• Next first difference = Last first difference + Next second difference
• Next first difference = $85 + 23 = 108$

Finally, we can find the next term in the original series:

• Next term = Last term in series + Next first difference
• Next term = $283 + 108 = 391$

Summary of Differences

Let's put this into a table to visualize the layers of differences:

The pattern of third differences (4, 2, 4, 2, 4) is consistent, leading us to 391 as the next term.

Conclusion

The number that replaces the question mark is 391.

Revision Table: Key Concepts for Number Series

Additional Information: Types of Number Series Patterns

Number series questions can involve various patterns. Some common types include:

• Arithmetic Series: Constant difference between terms.
• Geometric Series: Constant ratio between terms.
• Difference Series: The differences between terms follow a pattern (like in this problem, where the differences of differences have a pattern).
• Mixed Series: Combination of arithmetic, geometric, or other operations.
• Fibonacci-like Series: Each term is the sum of the previous two terms (or a variation).
• Prime Number Series: The terms are prime numbers or derived from them.
• Square/Cube Series: Terms are squares, cubes, or related to them (e.g., $n^2+1$, $n^3-1$).
• Alternating Series: Patterns involve alternating operations or sequences.

Analyzing the differences is a fundamental technique for many of these types, especially polynomial series or those with repeating patterns in the differences.