Solving the Number Series Question

The question asks us to find the number that replaces the question mark (?) in the given series: 1331, 2197, ?, 6859.

To solve this type of number series problem, we need to identify the underlying pattern or rule that relates the terms in the sequence.

Analyzing the Number Series Pattern

Let's look closely at the numbers provided: 1331, 2197, and 6859. These numbers are relatively large, which might indicate that they are powers of integers, such as squares or cubes.

Let's test if these numbers are perfect cubes by finding their cube roots:

• For the first term, 1331: We find that $\small 11 \times 11 \times 11 = 1331$. So, $\small 1331 = 11^3$.
• For the second term, 2197: We find that $\small 13 \times 13 \times 13 = 2197$. So, $\small 2197 = 13^3$.
• For the last term, 6859: We find that $\small 19 \times 19 \times 19 = 6859$. So, $\small 6859 = 19^3$.

Based on this analysis, the given number series appears to be composed of cubes of certain numbers:

$\small 11^3, 13^3, ?, 19^3$

Identifying the Pattern in the Bases

Now let's look at the bases of these cubes: 11, 13, and 19. We need to find the number that fits between 13 and 19 in this sequence of bases.

Consider the sequence of bases: 11, 13, ?, 19.

Let's examine the properties of these numbers:

• 11 is a prime number. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
• 13 is the next prime number immediately following 11.
• 19 is also a prime number.

This suggests that the pattern in the bases might be consecutive prime numbers.

Let's list the sequence of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, ...

Looking at our sequence of bases (11, 13, ?, 19) and comparing it with the list of prime numbers, we can see that the missing base is the prime number that comes after 13 and before 19.

The prime number that follows 13 is 17.

So, the missing base in the sequence 11, 13, ?, 19 is 17.

Calculating the Missing Number

Since the series consists of the cubes of these prime bases, the missing number is the cube of the missing base, which is 17.

We need to calculate $\small 17^3$:

$\small 17^3 = 17 \times 17 \times 17$

First, calculate $\small 17 \times 17$: $\small 17^2 = 289$.

Next, multiply 289 by 17:

$\small 289 \times 17 = 4913$

So, the missing number in the series is 4913.

Verifying the Solution

The completed series based on our pattern is: $\small 11^3, 13^3, 17^3, 19^3$.

Which translates to: 1331, 2197, 4913, 6859.

The calculated missing number is 4913. Let's check the given options:

• Option 1: 4913. This matches our result.
• Option 2: 5832. This is $\small 18^3$. The base 18 is not a prime number.
• Option 3: 4096. This is $\small 16^3$. The base 16 is not a prime number.
• Option 4: 3375. This is $\small 15^3$. The base 15 is not a prime number.

Our calculated number, 4913, matches one of the options and fits the pattern of cubes of consecutive prime numbers.

Summary of the Number Series Pattern

The pattern in the given series is that each term is the cube of a consecutive prime number, starting from 11.

The number that replaces the question mark (?) is 4913.

Revision Table: Key Number Series Patterns

Recognizing patterns is key to solving number series questions. Common patterns include:

• Addition or Subtraction Series (constant difference).
• Multiplication or Division Series (constant ratio).
• Square Series (terms are squares of numbers).
• Cube Series (terms are cubes of numbers).
• Series based on Prime Numbers.
• Series based on Composite Numbers.
• Fibonacci Series or similar recursive patterns.
• Combination of multiple patterns.

Always check for simple arithmetic or geometric progressions first, then look for squares, cubes, or patterns involving special numbers like primes.

Additional Information: Understanding Prime Numbers and Cubes

Prime Numbers: These are fundamental building blocks in number theory. A prime number has exactly two distinct positive divisors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, etc. Numbers greater than 1 that are not prime are called composite numbers.

Cubes: The cube of a number 'n' is 'n' multiplied by itself three times, denoted as $\small n^3$. For example, $\small 5^3 = 5 \times 5 \times 5 = 125$. Numbers that are the cube of an integer (like 1, 8, 27, 64, 125, 216, etc.) are called perfect cubes. This series uses perfect cubes where the base follows a specific pattern (prime numbers).

Solving number series often requires familiarity with basic arithmetic, powers (squares and cubes), and types of numbers (prime, composite, odd, even).