Consider the sequence 14, 916, 2536, ... where each term is formed by writing two square numbers side by side. What is the next term in this number pattern?

Introduction / Context:
This question involves a number sequence where each term is not a simple arithmetic or geometric progression, but a concatenation of two meaningful numbers. The challenge is to identify the hidden structure behind the sequence 14, 916, 2536 and then extend the pattern logically to find the next term. Such questions test pattern recognition, familiarity with squares of integers and the ability to think beyond simple addition or multiplication rules.

Given Data / Assumptions:

- The given sequence is 14, 916, 2536, ... - Each term appears to be formed by joining two numbers rather than normal arithmetic progression. - We are asked to find the next term in the same style. - There are four answer options, each a four digit number.

Concept / Approach:
The key observation is that each term looks like two perfect squares written side by side. For example, 14 can be seen as 1 and 4, which are 1^2 and 2^2. Similarly, 916 can be read as 9 and 16, which are 3^2 and 4^2, and 2536 is 25 and 36, which are 5^2 and 6^2. Once we recognize this pattern of consecutive square numbers, we simply continue with the next pair of consecutive squares and write them side by side to generate the next term in the sequence.

Step-by-Step Solution:
Step 1: Rewrite 14 as 1 and 4. These are 1^2 and 2^2, the squares of 1 and 2. Step 2: Rewrite 916 as 9 and 16. These are 3^2 and 4^2, the squares of 3 and 4. Step 3: Rewrite 2536 as 25 and 36. These are 5^2 and 6^2, the squares of 5 and 6. Step 4: Notice that the pattern uses consecutive pairs of integers: (1,2), (3,4), (5,6). Step 5: The next pair of consecutive integers after 5 and 6 is 7 and 8. Step 6: Compute their squares: 7^2 = 49 and 8^2 = 64. Step 7: Write these squares side by side without a separator to form the next term: 4964.

Verification / Alternative check:
To confirm, list the sequence in terms of square pairs: (1^2, 2^2) gives 14, (3^2, 4^2) gives 916, (5^2, 6^2) gives 2536, and the next should logically be (7^2, 8^2) which is 49 and 64, forming 4964. None of the other options match this continuation of consecutive square pairs. This confirmation shows that we correctly identified the underlying rule.

Why Other Options Are Wrong:
Options 4981, 6481 and 6449 do not correspond to any natural way of concatenating two consecutive perfect squares. There is no integer pair whose squares give (49,81), (64,81) or (64,49) in a consistent sequence following the earlier terms. Hence these options do not preserve the pattern of writing consecutive square numbers together.

Common Pitfalls:
A common mistake is to search for simple differences between terms or ratios, which do not reveal a consistent pattern here. Others may attempt to treat 14, 916 and 2536 as single numbers and miss the split into two parts. For sequence questions, it is often helpful to examine internal structure, such as splitting digits, looking at prime factors or considering known special numbers like squares and cubes.

Final Answer:
The next term in the sequence is 4964, corresponding to the concatenation of 7^2 and 8^2, which matches option C.