The number which comes after 36 in the series 4, 9, 16, 25, 36, ..... is

Concept: A series 4, 9, 16, 25, 36, ... in which each term is a perfect square is the sequence of squares of consecutive natural numbers: 22, 32, 42, 52, 62, and so on. Each term equals the square of the next natural number in order.

Application:

1. 22 = 4

2. 32 = 9

3. 42 = 16

4. 52 = 25

5. 62 = 36

6. The next natural number after 6 is 7, so the next term is 72 = 49.

Cross-check:

1. The differences between consecutive terms are 9 − 4 = 5, 16 − 9 = 7, 25 − 16 = 9, and 36 − 25 = 11 — consecutive odd numbers, each 2 more than the last.

2. Continuing this pattern, the next difference is 13, so the next term is 36 + 13 = 49, the same value obtained from the squares.

Result: The number that comes after 36 in the series is 49.