Understanding Number Series Patterns

Let's analyze the given number series to find the missing term: 131, 132, 141, 166, 215, ?, 417.

Solving number series questions involves identifying the underlying rule or pattern that connects the terms.

Finding the Pattern in the Series

To find the missing number in this number series, we examine the differences between consecutive terms.

• Difference between the 2nd term (132) and the 1st term (131): $132 - 131 = 1$
• Difference between the 3rd term (141) and the 2nd term (132): $141 - 132 = 9$
• Difference between the 4th term (166) and the 3rd term (141): $166 - 141 = 25$
• Difference between the 5th term (215) and the 4th term (166): $215 - 166 = 49$

Identifying the Difference Pattern

The sequence of differences we calculated is 1, 9, 25, 49. Let's look closely at these numbers.

We can observe that these differences are perfect squares:

• $1 = 1 \times 1 = 1^2$
• $9 = 3 \times 3 = 3^2$
• $25 = 5 \times 5 = 5^2$
• $49 = 7 \times 7 = 7^2$

The bases of these squares are 1, 3, 5, 7. These are consecutive odd numbers.

This suggests that the pattern in the number series is that the difference between consecutive terms is the square of consecutive odd numbers, starting with 1.

Calculating the Missing Number

Following the pattern of consecutive odd numbers (1, 3, 5, 7), the next odd number in the sequence is 9.

Therefore, the next difference in the number series should be $9^2$.

$9^2 = 9 \times 9 = 81$.

The missing number is found by adding this difference (81) to the last known term before the question mark, which is 215.

Missing number $= 215 + 81 = 296$.

Verifying the Number Series Pattern

To confirm that our identified pattern is correct, let's check the next step in the series. The odd number following 9 is 11.

According to the pattern, the difference between the term after the missing number (417) and the missing number (296) should be $11^2$.

$11^2 = 11 \times 11 = 121$.

Let's calculate the difference between 417 and 296:

$417 - 296 = 121$.

This result matches $11^2$, confirming that the pattern of adding squares of consecutive odd numbers ($1^2, 3^2, 5^2, 7^2, 9^2, 11^2$) is correct for this number series.

Summary of the Number Series Solution

The number series follows the rule where the difference between successive terms is the square of consecutive odd numbers starting from 1. The differences are $1^2, 3^2, 5^2, 7^2, 9^2, 11^2$.

The series progression is:

• 131
• $131 + 1^2 = 131 + 1 = 132$
• $132 + 3^2 = 132 + 9 = 141$
• $141 + 5^2 = 141 + 25 = 166$
• $166 + 7^2 = 166 + 49 = 215$
• $215 + 9^2 = 215 + 81 = 296$ (The missing number)
• $296 + 11^2 = 296 + 121 = 417$

Number Series Revision Table

Additional Information on Number Series

Number series problems are common in aptitude and logical reasoning tests. They assess your ability to quickly identify patterns and apply mathematical rules.

Key strategies for solving number series include:

• Calculating the differences between consecutive terms.
• Calculating the ratio between consecutive terms (for geometric series).
• Looking for patterns in the differences themselves (second-order differences).
• Checking for squares, cubes, or other powers.
• Identifying alternating patterns or combinations of rules.
• Considering prime numbers or other special number sequences.

Practice with various types of number series helps improve pattern recognition skills.