Understanding the Number Series Problem

This question asks us to find the missing number in a given series: 58, 60, 63, 68, 75, 86, ?. To solve number series problems like this, we need to identify the pattern or rule that connects the numbers in the sequence. This often involves looking at the differences between consecutive terms, ratios, or other mathematical operations.

Analyzing the Given Number Series

The given series is:

58, 60, 63, 68, 75, 86, ?

Let's examine the differences between consecutive terms to see if there is a recognizable pattern.

Calculating the Differences Between Terms

We subtract each term from the term that follows it:

• Difference between the 2nd and 1st term: $60 - 58 = 2$
• Difference between the 3rd and 2nd term: $63 - 60 = 3$
• Difference between the 4th and 3rd term: $68 - 63 = 5$
• Difference between the 5th and 4th term: $75 - 68 = 7$
• Difference between the 6th and 5th term: $86 - 75 = 11$

The sequence of differences is 2, 3, 5, 7, 11.

Identifying the Pattern in the Differences

Let's look at the sequence of differences: 2, 3, 5, 7, 11. These numbers are not consecutive integers, nor do they follow a simple arithmetic progression or geometric progression.

However, if we observe closely, these numbers (2, 3, 5, 7, 11) are the first few prime numbers in ascending order.

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

• The first prime number is 2.
• The second prime number is 3.
• The third prime number is 5.
• The fourth prime number is 7.
• The fifth prime number is 11.

This confirms that the differences between consecutive terms in the original series follow the sequence of prime numbers.

Determining the Next Number in the Series

Following the identified pattern, the next difference should be the next prime number after 11. The prime number immediately following 11 is 13.

To find the next term in the original series, we add this next difference (13) to the last term in the series (86).

Next term = Last term + Next difference

Next term = $86 + 13$

Next term = $99$

Therefore, the number that will replace the question mark (?) in the series is 99.

Summary of the Series and Differences

Conclusion

The series increases by consecutive prime numbers. The differences are 2, 3, 5, 7, 11. The next prime number is 13. Adding 13 to the last term, 86, gives $86 + 13 = 99$. The missing number is 99.

Revision Table: Number Series Concepts

Additional Information: Types of Number Series Patterns

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

• Arithmetic Progression: A constant difference between terms.
• Geometric Progression: A constant ratio between terms.
• Difference Series: The differences between terms form a pattern (like arithmetic, geometric, squares, cubes, prime numbers, etc.).
• Ratio Series: The ratios between terms form a pattern.
• Mixed Series: Involves a combination of two or more patterns or alternating patterns.
• Fibonacci Series: Each term is the sum of the two preceding terms (0, 1, 1, 2, 3, 5, ...).
• Square/Cube Series: Terms are squares or cubes of numbers, or based on them.

Solving number series problems requires careful observation and testing different potential patterns based on sums, differences, products, ratios, squares, cubes, and sequences of special numbers like prime numbers.