Understanding Number Series Patterns

This question asks us to find the missing number in the given series: 8, 14, ?, 44, 68, 98, 134. To solve number series questions, we need to identify the pattern or rule that governs the sequence of numbers.

Analyzing the Given Number Series

Let's look at the differences between consecutive terms in the series to see if we can find a pattern. The series is 8, 14, ?, 44, 68, 98, 134.

We calculate the difference between the known consecutive terms:

• Difference between 14 and 8: \(14 - 8 = 6\)
• Difference between 68 and 44: \(68 - 44 = 24\)
• Difference between 98 and 68: \(98 - 68 = 30\)
• Difference between 134 and 98: \(134 - 98 = 36\)

So, the sequence of differences we know is: 6, (difference between ? and 14), (difference between 44 and ?), 24, 30, 36.

Identifying the Pattern in Differences

Let's look at the differences we found: 6, ?, ?, 24, 30, 36. It appears there might be a pattern in these differences themselves.

Let's find the differences between these differences (the second level differences):

• Difference between 30 and 24: \(30 - 24 = 6\)
• Difference between 36 and 30: \(36 - 30 = 6\)

The second level differences are constant and equal to 6. This suggests that the first level differences are increasing by 6 each time.

Determining the Missing Number in the Series

Based on the pattern of the second level differences being 6, the sequence of first level differences should be:

• First difference: 6
• Second difference: \(6 + 6 = 12\)
• Third difference: \(12 + 6 = 18\)
• Fourth difference: \(18 + 6 = 24\) (Matches the series)
• Fifth difference: \(24 + 6 = 30\) (Matches the series)
• Sixth difference: \(30 + 6 = 36\) (Matches the series)

So the sequence of differences is 6, 12, 18, 24, 30, 36.

Now we can use these differences to find the missing number in the original series:

• The first term is 8.
• The second term is \(8 + 6 = 14\).
• The third term (the missing number) is \(14 + 12 = 26\).
• Let's check the next term: \(26 + 18 = 44\). This matches the given series.

Therefore, the missing number in the series is 26.

The missing number that replaces the question mark is 26.

Revision Table: Number Series

Additional Information on Solving Number Series

Solving number series problems often involves looking for various types of patterns. Here are some common ones:

• Arithmetic Progression: Each term is obtained by adding or subtracting a constant value to the previous term. The first differences are constant.
• Geometric Progression: Each term is obtained by multiplying or dividing the previous term by a constant value. Look at the ratios between consecutive terms.
• Arithmetic-Geometric Series: A combination of arithmetic and geometric progressions.
• Difference Series: The differences between consecutive terms form their own series which follows a recognizable pattern (like in this question, where the differences form an arithmetic series).
• Square/Cube Series: Terms are squares or cubes of natural numbers, or related to them (e.g., \(n^2\), \(n^2 \pm 1\), \(n^3\), \(n^3 \pm 1\)).
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 0, 1, 1, 2, 3, 5, 8...).
• Alternating Series: The pattern might involve operations that alternate (e.g., +3, -2, +3, -2...).
• Mixed Series: Two different series might be interleaved within one sequence. Look at alternate terms.

Systematic analysis, starting with simple differences, is often the best approach to crack number series problems.