Solving the Number Series Problem

The question asks us to find the number that should replace the question mark (?) in the given series: 339, 340, 348, ?, 439.

To solve this number series problem, we need to identify the pattern between consecutive terms.

Let's look at the differences between the given terms:

• Difference between the 2nd and 1st term: \(340 - 339 = 1\)
• Difference between the 3rd and 2nd term: \(348 - 340 = 8\)
• Let the missing term be \(x\). The difference between the 4th and 3rd term would be \(x - 348\).
• The difference between the 5th and 4th term would be \(439 - x\).

The differences we have found are 1 and 8. Let's see if there is a pattern in these differences.

• We notice that \(1 = 1^3\).
• We notice that \(8 = 2^3\).

This suggests a pattern where the difference between consecutive terms is the cube of consecutive natural numbers starting from 1.

Following this pattern, the next difference should be the cube of the next natural number, which is 3.

• The difference between the 4th term and the 3rd term should be \(3^3 = 27\).

So, the missing term (?) can be found by adding this difference to the 3rd term:

Missing Term = 3rd Term + \(3^3\)

Missing Term = \(348 + 27\)

Missing Term = \(375\)

Now, let's verify if this pattern holds for the last term in the series. The next difference should be the cube of 4.

• The difference between the 5th term and the 4th term (which we found to be 375) should be \(4^3 = 64\).

Let's check:

\(375 + 64 = 439\)

This matches the last term in the given series. Therefore, the pattern is consistent.

The pattern is adding the cubes of consecutive natural numbers (\(1^3, 2^3, 3^3, 4^3, \dots\)) to get the next term in the series.

The series can be represented as:

• \(339\)
• \(339 + 1^3 = 339 + 1 = 340\)
• \(340 + 2^3 = 340 + 8 = 348\)
• \(348 + 3^3 = 348 + 27 = 375\)
• \(375 + 4^3 = 375 + 64 = 439\)

Thus, the number that should come in place of the question mark (?) is 375.

Revision Table: Understanding Number Series

Additional Information on Number Series Patterns

Number series problems are common in logical reasoning and quantitative aptitude tests. Identifying the pattern is key to solving these problems. Common patterns include:

• Arithmetic Progression: The difference between consecutive terms is constant.
• Geometric Progression: The ratio between consecutive terms is constant.
• Difference Series: The differences between consecutive terms follow a pattern (e.g., they form an AP, GP, or a sequence of squares/cubes). This was the case in our problem where the differences were \(1^3, 2^3, 3^3, \dots\).
• Square Series: Terms are related to perfect squares (\(1^2, 2^2, 3^2, \dots\)) or differences are squares.
• Cube Series: Terms are related to perfect cubes (\(1^3, 2^3, 3^3, \dots\)) or differences are cubes. Our problem's pattern involved adding cubes.
• Mixed Series: A combination of different patterns (e.g., alternating addition and subtraction, or combination of arithmetic and geometric operations).
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 1, 1, 2, 3, 5, 8, ...).

To solve number series problems, try calculating differences between terms first. If the differences don't immediately show a simple pattern, try calculating the differences of the differences (second-order differences). Also, consider squares, cubes, prime numbers, or alternating patterns.