Finding the Missing Number in a Series

The question asks us to find the missing number in the given series: 11, 12, 20, 47, 111, ?.

To solve number series questions, we look for a pattern between consecutive terms. Let's examine the differences between the terms:

• Difference between the 2nd and 1st term: \(12 - 11 = 1\)
• Difference between the 3rd and 2nd term: \(20 - 12 = 8\)
• Difference between the 4th and 3rd term: \(47 - 20 = 27\)
• Difference between the 5th and 4th term: \(111 - 47 = 64\)

Let's list the differences we found:

\(1, 8, 27, 64, \dots\)

Now, let's try to identify the pattern in these differences. We can observe that these numbers are perfect cubes:

• \(1 = 1 \times 1 \times 1 = 1^3\)
• \(8 = 2 \times 2 \times 2 = 2^3\)
• \(27 = 3 \times 3 \times 3 = 3^3\)
• \(64 = 4 \times 4 \times 4 = 4^3\)

The pattern of the differences is the cube of consecutive natural numbers starting from 1. The differences are \(1^3, 2^3, 3^3, 4^3, \dots\).

Following this pattern, the next difference in the series should be \(5^3\).

Let's calculate the value of \(5^3\):

\(5^3 = 5 \times 5 \times 5 = 125\)

So, the missing number is obtained by adding this next difference (125) to the last given term (111).

Missing number \(= \text{Last term} + \text{Next difference}\)

Missing number \(= 111 + 125\)

Missing number \(= 236\)

Therefore, the missing number in the series is 236.

The complete series is 11, 12, 20, 47, 111, 236.

Breakdown of the Number Series Pattern

The missing number is \(111 + 125 = 236\).

Revision Table: Understanding Number Series

Additional Information on Solving Number Series Questions

Solving number series problems requires careful observation and pattern recognition. Here are some common strategies:

• Calculate Differences: Find the differences between consecutive terms. If the differences are constant, it's an arithmetic series. If not, look for a pattern in the differences themselves (first level difference, second level difference, etc.).
• Calculate Ratios: Find the ratio between consecutive terms. If the ratio is constant, it's a geometric series.
• Look for Squares and Cubes: Check if terms are perfect squares, cubes, or related to squares/cubes (e.g., \(n^2 \pm k\), \(n^3 \pm k\)).
• Identify Alternating Patterns: Sometimes, two different patterns alternate within the series.
• Look for Combined Operations: The pattern might involve a combination of arithmetic and multiplication/division operations.
• Prime Numbers: The series might involve prime numbers or operations related to them.
• Fibonacci-like Series: Each term might be the sum of the previous two terms (or a variation).

Practicing different types of number series helps in quickly identifying the underlying pattern.