In the series 169, 144, 121, 100, 85, 64, one term does not fit the underlying pattern. Identify the wrong term in this sequence.

Introduction / Context:
Here we are given a descending series of numbers and asked to find the term that breaks the pattern. Many such sequences are built from squares, cubes or other well known number sets. Recognizing a standard structure such as consecutive perfect squares makes it much easier to detect the intruder that does not belong.

Given Data / Assumptions:

• The series is: 169, 144, 121, 100, 85, 64.
• Exactly one term is incorrect with respect to a simple underlying rule.
• The other terms are assumed to follow a clean, mathematical pattern.
• The list is in strictly descending order apart from the anomaly.

Concept / Approach:
The numbers 169, 144, 121, 100 and 64 are all familiar because many of them are common perfect squares: 13^2, 12^2, 11^2, 10^2 and 8^2 respectively. This strongly suggests that the true series should be a descending run of square numbers. If so, the term between 100 and 64 should also be a perfect square. By comparing what we expect with the given 85, we can quickly identify the incorrect number.

Step-by-Step Solution:
Step 1: Express each known term as a square if possible. 169 = 13^2. 144 = 12^2. 121 = 11^2. 100 = 10^2. 64 = 8^2. Step 2: Note that these squares involve decreasing bases: 13, 12, 11, 10, (9), 8. Step 3: The missing square in this descending sequence should be 9^2 = 81, which should appear between 10^2 = 100 and 8^2 = 64. Step 4: The given term in that position is 85, which is not a perfect square and does not equal 81. Step 5: Therefore 85 is the inconsistent term and is the wrong number in the series.

Verification / Alternative check:
If we rewrite the corrected sequence as 169 (13^2), 144 (12^2), 121 (11^2), 100 (10^2), 81 (9^2), 64 (8^2), we clearly see a smooth pattern: the base of each square decreases by 1 at each step. The appearance of 85 breaks this otherwise perfect run of consecutive squares, confirming it as the outlier.

Why Other Options Are Wrong:
Each of the other candidates is a genuine perfect square: 169, 144, 121, 100, and 64 correspond to 13^2, 12^2, 11^2, 10^2 and 8^2. Removing any of these would destroy the clear and elegant structure of descending squares. Only 85 is not a square, so it is the only term that disrupts the pattern.

Common Pitfalls:
Sometimes students look at the differences between terms (for example 169 – 144 = 25, 144 – 121 = 23, etc.) and attempt to find patterns there, which are more complicated and less revealing. Instead, recognizing the base numbers that square to these values is faster and more reliable. Another pitfall is to assume that every series must have a constant difference, which is not true for square sequences.

Final Answer:
The only term that does not fit the descending sequence of perfect squares is 85.