Finding the Missing Number in the Series

The given series is 19, 26, ?, 46, 59, 74. We need to find the number that replaces the question mark (?). To solve number series problems, we look for a pattern in the numbers, usually by checking the differences or ratios between consecutive terms.

Analyzing the Series Pattern

Let's find the difference between consecutive terms where both numbers are known:

• Difference between the second and first term: $26 - 19 = 7$
• Difference between the fifth and fourth term: $59 - 46 = 13$
• Difference between the sixth and fifth term: $74 - 59 = 15$

So, the differences between terms are:

$19 \xrightarrow{+7} 26 \xrightarrow{+?} ? \xrightarrow{+?} 46 \xrightarrow{+13} 59 \xrightarrow{+15} 74$

The sequence of differences we know is 7, ?, ?, 13, 15. Let's look at the differences between these known differences:

• Difference between 15 and 13: $15 - 13 = 2$

This suggests that the differences between consecutive terms in the original series might be increasing by a constant value of 2. If this pattern holds, the differences should form the sequence 7, 9, 11, 13, 15.

Applying the Pattern to Find the Missing Number

Let's test this pattern by applying the differences 7, 9, 11, 13, 15 to the original series starting from the first term:

• First term: 19
• Second term: $19 + 7 = 26$ (Matches the given second term)
• Third term (the missing number): $26 + 9 = 35$
• Fourth term: $35 + 11 = 46$ (Matches the given fourth term)
• Fifth term: $46 + 13 = 59$ (Matches the given fifth term)
• Sixth term: $59 + 15 = 74$ (Matches the given sixth term)

The pattern of adding consecutive odd numbers (7, 9, 11, 13, 15) starting from 7 perfectly fits the given series. Therefore, the missing number is 35.

Conclusion

The number that replaces the question mark (?) in the series 19, 26, ?, 46, 59, 74 is 35. The pattern involves adding differences that increase by 2 each time (specifically, adding consecutive odd numbers starting from 7).

Revision Table: Number Series Analysis

Additional Information: Types of Number Series Patterns

Number series questions often follow various patterns. Understanding common types can help in solving them quickly:

• Arithmetic Series: A constant difference is added or subtracted between consecutive terms (e.g., 2, 4, 6, 8...).
• Geometric Series: A constant ratio is multiplied or divided between consecutive terms (e.g., 3, 6, 12, 24...).
• Difference Series: The pattern is in the differences between consecutive terms. The differences themselves might form an arithmetic series (as in this question), a geometric series, or another pattern.
• Mixed Series: Combinations of different operations (addition, subtraction, multiplication, division) or alternating patterns.
• Square/Cube Series: Terms are related to squares or cubes of numbers (e.g., $1^2, 2^2, 3^2...$ or $1^3, 2^3, 3^3...$).
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 1, 1, 2, 3, 5, 8...).

Identifying the type of pattern is the key to solving number series problems. Checking differences is often the first step to uncover arithmetic or difference series patterns.