Solving the Number Series Puzzle

Let's analyze the given number series to find the pattern and determine the number that replaces the question mark (?). The series is: 116, 128, 146, 170, 200, ?

To find the pattern in a number series, we often look at the differences between consecutive terms.

Calculating the Differences Between Terms

We calculate the difference between each adjacent pair of numbers in the series:

• Difference between 128 and 116: $128 - 116 = 12$
• Difference between 146 and 128: $146 - 128 = 18$
• Difference between 170 and 146: $170 - 146 = 24$
• Difference between 200 and 170: $200 - 170 = 30$

So, the sequence of differences is: 12, 18, 24, 30.

Identifying the Pattern in the Differences

Now, let's look at the differences we just calculated (12, 18, 24, 30). Is there a pattern here?

Let's find the difference between consecutive terms in this new sequence (the differences of the original series):

• Difference between 18 and 12: $18 - 12 = 6$
• Difference between 24 and 18: $24 - 18 = 6$
• Difference between 30 and 24: $30 - 24 = 6$

The differences between the consecutive terms are constant, always 6. This indicates that the first differences (12, 18, 24, 30) form an arithmetic progression with a common difference of 6.

Finding the Next Term in the Series

Since the differences are increasing by 6 each time, the next difference in the sequence (12, 18, 24, 30, ?) should be $30 + 6 = 36$.

To find the number that replaces the question mark in the original series, we need to add this next difference (36) to the last number in the series (200).

The next number is $200 + 36 = 236$.

Conclusion

The pattern involves adding a number that increases by 6 each time. Following this pattern, the number that replaces the question mark (?) is 236.

The series with the missing number filled in is: 116, 128, 146, 170, 200, 236.

Revision Table: Understanding Number Series

Solving number series questions often involves identifying underlying patterns. Common patterns include:

• Arithmetic Progression: Adding or subtracting a constant value.
• Geometric Progression: Multiplying or dividing by a constant value.
• Differences: Finding the difference between consecutive terms. Sometimes the differences themselves form a pattern (like in this case, an arithmetic progression).
• Squares or Cubes: Terms might be squares, cubes, or related to them ($n^2, n^3, n^2+1$).
• Fibonacci-like Series: The next term is the sum of the previous two terms.
• Alternating Patterns: Different patterns apply to alternate terms.

Additional Information: Second-Order Series

The number series in this question is an example of a second-order arithmetic series. This means that the difference between consecutive terms does not stay constant, but the difference of those differences (the second difference) is constant. Identifying this 'difference of differences' pattern is a key technique for solving such series. When you find the first differences, analyze them as a new series. If *that* series has a simple pattern (like an arithmetic or geometric progression), you can predict the next first difference and thus the next term in the original series.