To determine the number that should fill the empty cage in the sequence 20, 12, 10, 11, _, 6.75, we need to identify the pattern in the given numbers.
Let's carefully analyze the differences between successive numbers:
1. The difference between 20 and 12:
[tex]\[
20 - 12 = 8
\][/tex]
2. The difference between 12 and 10:
[tex]\[
12 - 10 = 2
\][/tex]
3. The difference between 10 and 11:
[tex]\[
10 - 11 = -1
\][/tex]
4. We need to determine the difference for the empty cage and then check it against the difference to 6.75:
[tex]\[
\text{Let the number be } X.
\][/tex]
[tex]\[
\text{The difference needed from } 11 \text{ to } X \text{ should continue the pattern of differences (8, 2, -1,...)}
\][/tex]
Based on the previous numbers, let's assume the differences continue to decrease in a pattern:
[tex]\[
8 \rightarrow 2 \rightarrow -1 \rightarrow ???
\][/tex]
Observing the pattern more clearly:
- The decrement from 8 to 2 is 6.
- The decrement from 2 to -1 is 3.
Following this observation, we see that the next decrement might continue by 3 to follow the pattern:
[tex]\[ -1 - 4 = -5 \][/tex]
Hence, we need to calculate:
[tex]\[
11 + (-5) = 6
\][/tex]
Thus, the number that should fill the empty cage is:
[tex]\[
\boxed{6}
\][/tex]
This resultant number matches the progressive pattern of decreasing differences. Therefore, the most appropriate number to fit the given pattern is:
6
