Analyzing the Number Series Pattern

The question asks us to find the next number in the given series: 382, 322, 272, 232, 202, ?. To solve this number series puzzle, we need to identify the underlying pattern or rule that governs the sequence of numbers. Let's examine the differences between consecutive terms in the series.

Step-by-Step Number Series Solution

We calculate the difference between each pair of adjacent numbers:

• Difference between the 1st and 2nd term: \(382 - 322 = 60\)
• Difference between the 2nd and 3rd term: \(322 - 272 = 50\)
• Difference between the 3rd and 4th term: \(272 - 232 = 40\)
• Difference between the 4th and 5th term: \(232 - 202 = 30\)

Let's look at these differences: 60, 50, 40, 30. We can see a clear pattern here. The differences are decreasing by 10 each time.

• \(60 - 10 = 50\)
• \(50 - 10 = 40\)
• \(40 - 10 = 30\)

Following this pattern of decreasing differences, the next difference in the series should be \(30 - 10 = 20\).

To find the next number in the original series, we subtract this predicted difference (20) from the last known term (202).

Next term = Last term - Next difference

Next term = \(202 - 20 = 182\)

Therefore, the number that replaces the question mark (?) in the series is 182.

Summarizing the Number Series Pattern

We can represent the pattern observed in the series and its differences in a table:

The pattern in the differences (60, 50, 40, 30, 20) is an arithmetic progression with a common difference of -10. This confirms that the next number is indeed 182.

Identifying the Correct Option

Based on our calculation, the next number in the series is 182. Let's check the given options:

• 168
• 150
• 182
• 132

The calculated number, 182, matches one of the options.

Final Answer Determination

The number that completes the given number series 382, 322, 272, 232, 202, ? is 182.

Revision Table: Number Series Analysis

Additional Information: Solving Number Series Problems

Solving number series problems often involves looking for various types of patterns:

• Arithmetic Series: The difference between consecutive terms is constant.
• Geometric Series: The ratio between consecutive terms is constant.
• Difference Series: The differences between consecutive terms themselves form a pattern (like in this problem, the differences formed an arithmetic series).
• Mixed Series: Combinations of arithmetic, geometric, or other operations.
• Fibonacci-like Series: Each term is the sum of the previous two (or more) terms.
• Square or Cube Series: Terms are related to squares or cubes of numbers, possibly with additions or subtractions.

It is helpful to calculate the differences between consecutive terms as a first step, as this often reveals the underlying pattern, as seen in this series problem.