Number Series: Identify Wrong Number

The task is to examine the given number series: 1, 4, 9, 16, 26, 36, and pinpoint the single number that does not follow the established pattern.

Series Pattern Analysis

To find the incorrect number, we first need to understand the pattern governing the sequence. Let's explore the relationship between the numbers.

Differences Between Series Terms

Calculating the difference between consecutive terms can reveal a pattern:

• \(4 - 1 = 3\)
• \(9 - 4 = 5\)
• \(16 - 9 = 7\)
• \(26 - 16 = 10\)
• \(36 - 26 = 10\)

The differences observed are \(3, 5, 7, 10, 10\). The initial differences (\(3, 5, 7\)) suggest a sequence of increasing odd numbers. If this pattern continued, the next difference should be \(9\). However, the series shows a difference of \(10\), followed by another \(10\). This deviation indicates a potential issue.

Square Numbers Pattern in Series

Let's investigate if the numbers correspond to squares of natural numbers:

• The first number is \(1\), which is equal to \(1^2\).
• The second number is \(4\), which is equal to \(2^2\).
• The third number is \(9\), which is equal to \(3^2\).
• The fourth number is \(16\), which is equal to \(4^2\).

This observation strongly suggests that the series is intended to be a sequence of consecutive squares.

Calculating Expected Series Numbers

Based on the identified pattern of squares (\(n^2\)), we can predict the expected numbers in the series:

• The fifth term should be \(5^2\), which equals \(25\).
• The sixth term should be \(6^2\), which equals \(36\).

The provided series is \(1, 4, 9, 16, 26, 36\). Comparing this with the expected sequence \(1, 4, 9, 16, 25, 36\), it is clear that the fifth term, \(26\), deviates from the pattern.

Identifying the Incorrect Number

The number \(26\) in the series should logically be \(25\) to maintain the consistent pattern of squares (\(n^2\)) where \(n\) increases by one for each subsequent term. Therefore, \(26\) is the incorrect number.

Series Conclusion

The number series follows the pattern of squares of natural numbers (\(1^2, 2^2, 3^2, \dots\)). The number \(26\) disrupts this pattern and should be \(25\) (\(5^2\)).