Consider the number series where each term is formed by concatenating two consecutive perfect squares: 14, 916, 2536, .... What is the next term in this series?

Introduction:
This is a pattern-recognition number series question. Instead of a simple arithmetic or geometric progression, each term is constructed from two familiar mathematical objects: perfect squares. You must identify the rule and then extend the pattern to find the next term.

Given Data / Assumptions:

• Given series: 14, 916, 2536, ...
• Each term appears to be more than a simple single number; it looks like a combination of two smaller numbers.
• We need to determine the next term after 2536.

Concept / Approach:
Look carefully at each term and try to break it into two meaningful parts. The first term “14” can be seen as “1” and “4”. The second term “916” can be seen as “9” and “16”. The third term “2536” can be seen as “25” and “36”. These are all perfect squares of consecutive integers: 1^2, 2^2, 3^2, 4^2, 5^2, 6^2.

Step-by-Step Solution:
Step 1: Break each term into two parts.14 → 1 and 4916 → 9 and 162536 → 25 and 36Step 2: Recognize the squares.1 = 1^2, 4 = 2^29 = 3^2, 16 = 4^225 = 5^2, 36 = 6^2Step 3: Spot the pattern.Each term is formed by concatenating two consecutive perfect squares.Term 1: 1^2 and 2^2 → 1, 4 → 14Term 2: 3^2 and 4^2 → 9, 16 → 916Term 3: 5^2 and 6^2 → 25, 36 → 2536Step 4: Find the next pair of squares.Next consecutive squares are 7^2 = 49 and 8^2 = 64.Concatenate 49 and 64 → 4964

Verification / Alternative check:
We have used consecutive squares: (1,2), (3,4), (5,6), (7,8). At each stage, the concatenation rule is consistent. This confirms that 4964 follows logically as the next term.

Why Other Options Are Wrong:
4981, 6481 and 6449 do not represent concatenations of two consecutive perfect squares. 36100 is unrelated to the observed structure. Only 4964 matches 7^2 and 8^2 placed side by side.

Common Pitfalls:
Students often look for differences between terms (916 − 14, 2536 − 916, etc.) and miss the structural pattern inside each term. Whenever numbers look “glued together,” always consider concatenation of known sequences like squares, cubes or primes.

Final Answer:
The next number in the series is 4964.