Finding the Missing Number in the Number Series

The question asks us to find the number that replaces the question mark (?) in the given number series: 55, 59, 68, 84, 109, ?

To solve this number series problem, we need to identify the pattern or rule that governs the sequence of numbers. Let's look at the differences between consecutive terms in the series.

Step-by-Step Analysis of the Number Series Pattern

1. Calculate the difference between the first two terms: $$59 - 55 = 4$$
2. Calculate the difference between the second and third terms: $$68 - 59 = 9$$
3. Calculate the difference between the third and fourth terms: $$84 - 68 = 16$$
4. Calculate the difference between the fourth and fifth terms: $$109 - 84 = 25$$

Let's list the differences we found between consecutive terms:

• Difference 1 (between 55 and 59): 4
• Difference 2 (between 59 and 68): 9
• Difference 3 (between 68 and 84): 16
• Difference 4 (between 84 and 109): 25

Identifying the Pattern in the Differences

Now let's examine the sequence of differences: 4, 9, 16, 25. Do you notice a pattern here? These numbers are perfect squares:

• $4 = 2^2$
• $9 = 3^2$
• $16 = 4^2$
• $25 = 5^2$

The pattern in the differences is the square of consecutive integers starting from 2 ($2^2, 3^2, 4^2, 5^2$).

Predicting the Next Difference and Term

Following this pattern, the next difference in the number series should be the square of the next integer, which is 6. So, the next difference is $6^2$.

$$6^2 = 36$$

To find the next number in the series (the one that replaces the question mark), we need to add this next difference (36) to the last term in the series (109).

$$109 + 36 = 145$$

Therefore, the number that replaces the question mark is 145.

Summary of the Number Series Pattern

The number series follows a pattern where each term is obtained by adding the square of a consecutive integer (starting from 2) to the previous term.

The number that replaces the question mark is 145.

Revision Table: Number Series Concepts

Additional Information: Solving Number Series Questions

Solving number series questions is a common part of reasoning and quantitative aptitude tests. Here are some tips:

• Calculate Differences: Always start by finding the difference between consecutive terms. If the differences show a pattern, you've likely found the rule.
• Look for Ratios: If differences don't reveal a simple pattern, check for ratios between consecutive terms (e.g., if the series involves multiplication or division).
• Check for Squares/Cubes: Many number series patterns involve squares ($1, 4, 9, 16, ...$) or cubes ($1, 8, 27, 64, ...$).
• Consider Alternating Patterns: Sometimes, two different patterns might be interleaved in the same series.
• Combine Operations: The pattern might involve a combination of operations, like multiply by a number and then add/subtract another number.
• Prime Numbers: Look for sequences involving prime numbers (2, 3, 5, 7, 11, ...).
• Practice: The more number series problems you practice, the better you become at recognizing different types of patterns quickly.