In the descending sequence of perfect squares 196, 169, 144, 121, 100, 81, what number should come next?

Introduction / Context:
This problem checks your understanding of perfect squares and how they appear in number series. Each term here corresponds to the square of a natural number. Recognizing such square patterns is a core skill in aptitude questions dealing with sequences and series.

Given Data / Assumptions:

• Sequence given: 196, 169, 144, 121, 100, 81, ?
• The numbers are arranged in decreasing order.
• We assume the sequence is built from perfect squares of consecutive integers.

Concept / Approach:
We recall that perfect squares are numbers of the form n^2 where n is a positive integer. The key idea is to express each term as a square and see how the base integers change. Once we identify the pattern in those base integers, predicting the next term is straightforward.

Step-by-Step Solution:
196 can be written as 14 * 14, so 196 = 14^2.169 can be written as 13 * 13, so 169 = 13^2.144 can be written as 12 * 12, so 144 = 12^2.121 can be written as 11 * 11, so 121 = 11^2.100 can be written as 10 * 10, so 100 = 10^2.81 can be written as 9 * 9, so 81 = 9^2.We clearly see that the base integers are decreasing consecutively: 14, 13, 12, 11, 10, 9. Therefore, the next base integer is 8, and the next term is 8^2 = 64.

Verification / Alternative check:
If we continue the sequence of bases as 14, 13, 12, 11, 10, 9, 8 and square each, we get 196, 169, 144, 121, 100, 81 and 64, which is consistent. None of the other options produce a square that fits the pattern of consecutive integers. For example, 77, 74 and 67 are not perfect squares, while 49 would correspond to 7^2 which is the term after 64, not the immediate next one in this list.

Why Other Options Are Wrong:
77, 74 and 67 are not perfect squares of integers, so they cannot continue a sequence clearly built from perfect squares.

49 is a perfect square (7^2), but it would be the term after 64 in the descending chain 14^2, 13^2, ..., 9^2, 8^2, 7^2. Since we only need the immediate next term, 49 is too far ahead.

Common Pitfalls:
Some students try to find differences between terms (27, 25, 23, 21, 19) and may get distracted. The differences themselves are decreasing by 2, which is another hint that these are squares of consecutive integers. However, focusing on the square structure is cleaner and less error prone. Forgetting standard squares such as 196, 169 or 144 can also slow you down.

Final Answer:
The next number in the sequence is 64.