Analysing the Number Series Problem

The question asks us to find the missing number in the given number series: 11, 36, ?, 121, 185, 266.

To solve number series problems, we usually look for a pattern between consecutive terms. This pattern could involve addition, subtraction, multiplication, division, squares, cubes, or a combination of these operations.

Step-by-Step Solution to Find the Missing Number

Let's examine the differences between the consecutive terms where we know both numbers:

• Difference between the second and first term: $36 - 11 = 25$
• Difference between the fifth and fourth term: $185 - 121 = 64$
• Difference between the sixth and fifth term: $266 - 185 = 81$

The differences we have found are 25, 64, and 81. Let's look closely at these numbers to see if they form a pattern:

• $25 = 5^2$
• $64 = 8^2$
• $81 = 9^2$

We see that these differences are squares of numbers. The bases of the squares are 5, 8, and 9.

The differences occur between:

• Term 1 and Term 2: $5^2$
• Term 4 and Term 5: $8^2$
• Term 5 and Term 6: $9^2$

The pattern of the bases (5, ?, ?, 8, 9) seems to be a sequence of consecutive integers starting from 5. Let's assume the bases are 5, 6, 7, 8, 9.

If this is the case, the differences between consecutive terms should be the squares of these numbers: $5^2, 6^2, 7^2, 8^2, 9^2$.

These differences would be 25, 36, 49, 64, 81.

Verifying the Number Series Pattern

Let's test this pattern with the given series starting from the first term:

• First term: 11
• Second term: $11 + 5^2 = 11 + 25 = 36$. This matches the given second term.
• Third term (the missing number): $36 + 6^2 = 36 + 36 = 72$.
• Fourth term: $72 + 7^2 = 72 + 49 = 121$. This matches the given fourth term.
• Fifth term: $121 + 8^2 = 121 + 64 = 185$. This matches the given fifth term.
• Sixth term: $185 + 9^2 = 185 + 81 = 266$. This matches the given sixth term.

The pattern fits perfectly. The missing number is 72.

Summary of the Number Series and Pattern

The missing number in the series is 72.

Revision Table for Number Series Patterns

Here are some common types of number series patterns you might encounter in aptitude questions:

• Arithmetic Series: The difference between consecutive terms is constant (e.g., 2, 5, 8, 11...).
• Geometric Series: Each term is found by multiplying the previous term by a constant ratio (e.g., 3, 6, 12, 24...).
• Difference Series: The differences between consecutive terms form a pattern (e.g., increasing by a constant, squares, cubes, prime numbers). This is the type of pattern found in this problem.
• Double Difference Series: The differences between the differences form a pattern.
• Squares or Cubes Series: Terms are squares or cubes, sometimes with an addition or subtraction (e.g., $1^2+1, 2^2+1, 3^2+1...$ or $1^3, 2^3, 3^3...$).
• Fibonacci Series: Each term is the sum of the two preceding ones (e.g., 0, 1, 1, 2, 3, 5...).
• Mixed Series: A combination of two or more patterns.

Additional Information on Solving Number Series

Solving number series problems requires careful observation and trying out different possible patterns. Here are some tips:

• Calculate the differences between consecutive terms first. If there's no clear pattern, calculate the differences of these differences.
• Look for ratios if the terms are increasing or decreasing rapidly, suggesting multiplication or division.
• Check if the terms are related to squares, cubes, or prime numbers.
• Sometimes, the pattern might alternate between two different operations.
• Practicing various types of series helps in quickly recognizing patterns.