To solve the problem of finding the missing number in the arithmetic sequence [tex]\(20, \ldots, 36, 44\)[/tex], we first need to determine if there is a common difference between the terms, which is characteristic of an arithmetic sequence.
Consider the given numbers: [tex]\(20\)[/tex], [tex]\(36\)[/tex], and [tex]\(44\)[/tex]. We need to find the common differences between consecutive terms:
1. Calculate the difference between the second term and the first term:
[tex]\[
36 - 20 = 16
\][/tex]
2. Calculate the difference between the third term and the second term:
[tex]\[
44 - 36 = 8
\][/tex]
For an arithmetic sequence, the common difference should be the same between all consecutive terms. Here, the differences are not the same (16 and 8).
To check if a number could potentially be missing, let's reasonably assume the sequence could be:
[tex]\[
20, \ldots, 28, 36, 44
\][/tex]
Let's insert 28 as the potential missing number and check if it forms a valid arithmetic sequence:
3. Calculate the difference from 20 to 28:
[tex]\[
28 - 20 = 8
\][/tex]
4. Calculate the difference from 28 to 36:
[tex]\[
36 - 28 = 8
\][/tex]
Therefore, by inserting 28, we have the sequence:
[tex]\[
20, 28, 36, 44
\][/tex]
The differences between consecutive terms are consistent (both are 8), which indeed forms an arithmetic sequence.
Thus, the missing number in the arithmetic sequence [tex]\(20, \ldots, 36, 44\)[/tex] is [tex]\(28\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{28}
\][/tex]
