To determine the next number in the sequence \(\{21, 20, 18, 15, 11\}\), we first need to analyze the pattern of the sequence. We will look at the differences between consecutive terms to see if there is a pattern.
Let's calculate the differences:
1. Difference between \(21\) and \(20\):
[tex]\[
21 - 20 = 1
\][/tex]
2. Difference between \(20\) and \(18\):
[tex]\[
20 - 18 = 2
\][/tex]
3. Difference between \(18\) and \(15\):
[tex]\[
18 - 15 = 3
\][/tex]
4. Difference between \(15\) and \(11\):
[tex]\[
15 - 11 = 4
\][/tex]
So, the differences are \(1, 2, 3, 4\).
We notice that the differences are increasing by \(1\) each time:
[tex]\[
1, 2, 3, 4
\][/tex]
Following this pattern, the next difference should be:
[tex]\[
4 + 1 = 5
\][/tex]
To find the next number in the sequence, we subtract this new difference from the last number in the sequence (\(11\)):
[tex]\[
11 - 5 = 6
\][/tex]
Therefore, the next number in the sequence is:
[tex]\[
\boxed{6}
\][/tex]
