Understanding Number Series Patterns

This question asks us to find the missing number in a given number series: 8, 15, 26, ?, 56, 75. To solve this type of number series problem, we need to identify the pattern or rule that governs the sequence of numbers. Often, this involves looking at the differences between consecutive terms, the ratio between terms, or a combination of operations.

Step-by-Step Analysis of the Series

Let's examine the differences between the consecutive terms in the given number series:

• Difference between the second and first term: $15 - 8 = 7$
• Difference between the third and second term: $26 - 15 = 11$
• Difference between the sixth and fifth term: $75 - 56 = 19$

The differences we have found are 7, 11, and 19. Let the missing term be $x$. The differences involving the missing term would be $x - 26$ and $56 - x$. So the sequence of differences is 7, 11, $(x - 26)$, $(56 - x)$, 19.

Identifying the Pattern in the Differences

Let's look at the differences between the known differences: $11 - 7 = 4$. This suggests a possible pattern in the increases of the differences. Let's test the options provided for the missing term (?) to see if a consistent pattern emerges.

Consider the options:

• Option 1: 39
• Option 2: 41
• Option 3: 35
• Option 4: 43

Let's assume the missing term is 39 (Option 1) and calculate the differences between consecutive terms:

• $15 - 8 = 7$
• $26 - 15 = 11$
• $39 - 26 = 13$
• $56 - 39 = 17$
• $75 - 56 = 19$

The sequence of differences is 7, 11, 13, 17, 19.

Now, let's look at the differences between these differences (the second level of differences):

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

The second level of differences is 4, 2, 4, 2. This shows a clear repeating pattern of alternating increases of 4 and 2 in the first-level differences.

Verifying the Pattern with the Proposed Missing Number

If we use the pattern of first-level differences (7, 11, 13, 17, 19) derived assuming 39 is the missing number, we can reconstruct the original series:

• Start with the first term: 8
• Add the first difference: $8 + 7 = 15$ (Second term)
• Add the second difference: $15 + 11 = 26$ (Third term)
• Add the third difference: $26 + 13 = 39$ (Fourth term - missing number)
• Add the fourth difference: $39 + 17 = 56$ (Fifth term)
• Add the fifth difference: $56 + 19 = 75$ (Sixth term)

The reconstructed series is 8, 15, 26, 39, 56, 75, which perfectly matches the given series when the missing number is 39.

Let's summarize this in a table:

As shown in the table, the pattern of adding alternating increments of 4 and 2 to the differences holds true throughout the series when the missing term is 39.

Conclusion

The missing number in the series 8, 15, 26, ?, 56, 75 is 39 because it fits the pattern where the differences between consecutive terms increase alternately by 4 and 2 (7, 11, 13, 17, 19).

Revision Table: Number Series Pattern

Additional Information: Types of Number Series Patterns

Number series problems can have various patterns. Some common types include:

• Arithmetic Series: The difference between consecutive terms is constant.
• Geometric Series: Each term is multiplied by a constant ratio to get the next term.
• Difference Series: The differences between consecutive terms follow a pattern (like an arithmetic series, geometric series, or the alternating pattern seen in this question).
• Double Difference Series: The differences between the differences follow a pattern.
• Alternating Series: Terms alternate between increasing and decreasing, or involve alternating operations.
• Square or Cube Series: Terms are related to squares or cubes of natural numbers, possibly with additions or subtractions.
• Mixed Series: A combination of two or more patterns within a single series.
• Fibonacci Series related: Each term is the sum of the two preceding terms, or variations of this.

Solving number series requires careful observation and testing different potential patterns.