Using sequence differences, you have this:
7, 12, 20, 31
5,  8,  11
3,  3,  3
From the above, we can formulate a "closed form" as follows:
a_n = 1/2 (3 n^2 - 5 n + 12), which will give you each term in your sequence as follows:
5, 7, 12, 20, 31, 45, 62, 82, 105, 131, 160, 192, 227, 265, 306, 350, 397, 447, 500, ....etc.

The given sequence is 7, 12, 20, 31, __.

To find the missing number in the sequence, let's analyze the pattern and the relationship between the given numbers.

Identifying the pattern:

Upon observing the given sequence, we can see that the numbers are increasing with each term. However, the rate of increase is not constant. Let's examine the differences between consecutive terms to understand the pattern better.

The differences between consecutive terms are as follows:
12 - 7 = 5
20 - 12 = 8
31 - 20 = 11

Identifying the relationship:

By observing the differences, we can notice that the differences between consecutive terms are increasing by 3 each time.

Forming a formula:

To find the missing number, we can use this pattern and form a formula to calculate the next term.

Let's denote the given sequence as S and the position of the term as n.

The formula to find the nth term of the sequence can be written as:
S(n) = S(n-1) + (n+1)

Calculating the missing number:

To find the missing number, we need to calculate the 5th term of the sequence using the formula.

S(5) = S(4) + (5+1)

S(4) = 31 (last given term)

S(5) = 31 + (5+1)
= 31 + 6
= 37

Therefore, the missing number in the given sequence is 37.

Summary:

The missing number in the sequence 7, 12, 20, 31, __ is 37. The pattern in the sequence is that each term is obtained by adding the position of the term to the previous term. By using this pattern, we can calculate the missing number as the 5th term of the sequence.