To solve this sequence, let's look for a pattern or rule that governs the relationship between the numbers.
Looking at the given sequence: 1, 2, 4, 13, 31, 112, ?
We can observe that each number in the sequence is obtained by manipulating the previous number in some way.
Let's examine the differences between consecutive terms:
2 - 1 = 1
4 - 2 = 2
13 - 4 = 9
31 - 13 = 18
112 - 31 = 81
The differences between consecutive terms are not constant. Therefore, the sequence does not follow a simple arithmetic progression or geometric progression.
Let's try looking at the differences between the differences:
2 - 1 = 1
9 - 2 = 7
18 - 9 = 9
81 - 18 = 63
The second differences are not constant either. So, it doesn't seem to be a quadratic sequence.
However, if we observe closely, we can notice that the sequence is formed by adding the square of a number to the previous term. Let's break it down:
1 + 1^2 = 2
2 + 2^2 = 4
4 + 3^2 = 13
13 + 4^2 = 31
31 + 5^2 = 112
Following this pattern, the next term in the sequence would be:
112 + 6^2 = 112 + 36 = 148
Therefore, the missing term in the sequence is 148. However, none of the given options matches this value.
Hence, the correct answer is D) none.