Solving the Number Series Problem

This question asks us to find the missing number in a given series: 123, 194, 267, ?, 429. To solve number series problems, we need to identify the pattern or rule that connects consecutive terms.

Identifying the Pattern in the Number Series

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

• Difference between the second and first term: \$194 - 123 = 71\$
• Difference between the third and second term: \$267 - 194 = 73\$

The first two differences we found are 71 and 73. Let's examine these differences to see if they follow a pattern.

We notice that 71 and 73 are consecutive prime numbers. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Let's test the hypothesis that the pattern is adding consecutive prime numbers starting from 71.

Applying the Prime Number Pattern

If the pattern is to add consecutive prime numbers, the next prime number after 73 is 79. Let's add 79 to the last known term (267) to find the missing number.

Missing number \$= 267 + 79\$

Missing number \$= 346\$

Now, let's verify this with the next term in the series. The prime number after 79 is 83. If our missing number is correct (346), adding the next prime number (83) should give us the last number in the series (429).

Check: \$346 + 83 = 429\$

This matches the last number provided in the series. Therefore, the pattern is confirmed: add consecutive prime numbers starting from 71 to get the next term in the series.

Step-by-Step Solution

1. Find the difference between the first two terms: \$194 - 123 = 71\$.
2. Find the difference between the second and third terms: \$267 - 194 = 73\$.
3. Observe that 71 and 73 are consecutive prime numbers.
4. Identify the next prime number after 73, which is 79.
5. Add this prime number (79) to the last known term (267) to find the missing term: \$267 + 79 = 346\$.
6. Identify the next prime number after 79, which is 83.
7. Verify the pattern by adding 83 to the calculated missing term (346) to see if it matches the final term (429): \$346 + 83 = 429\$.
8. Since it matches, the missing number in the series is 346.

Summary of the Pattern

The missing number in the series is 346.

Revision Table: Number Series and Patterns

Additional Information: Types of Number Series Patterns

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

• Arithmetic Series: Each term is obtained by adding a constant value to the previous term (constant difference).
• Geometric Series: Each term is obtained by multiplying the previous term by a constant value (constant ratio).
• Difference Series: The differences between consecutive terms follow a pattern (e.g., constant difference, arithmetic series, prime numbers, squares, cubes). This is the type seen in this problem.
• Mixed Series: A combination of two or more patterns or alternating patterns.
• Fibonacci Series: Each term is the sum of the two preceding terms (starting usually with 0 and 1 or 1 and 1).
• Series based on Squares or Cubes: Terms are related to squares or cubes of natural numbers.

Solving number series problems requires careful observation and systematic testing of potential patterns, often starting by looking at the differences or ratios between terms.