To identify the next three terms of the given sequence [tex]$10, 6, 2, \ldots$[/tex], we need to determine the pattern governing the sequence.
First, observe that the difference between consecutive terms is consistent:
[tex]\[
6 - 10 = -4
\][/tex]
[tex]\[
2 - 6 = -4
\][/tex]
This indicates that the sequence follows an arithmetic pattern where each term is decreased by 4 from the previous term.
Given this pattern, let's find the next three terms:
1. Start with the last known term in the sequence, which is 2.
2. Subtract 4 from 2 to find the first new term:
[tex]\[
2 - 4 = -2
\][/tex]
So, the next term after 2 is [tex]$-2$[/tex].
3. Now use [tex]$-2$[/tex] as the new starting term and subtract 4 again to find the second new term:
[tex]\[
-2 - 4 = -6
\][/tex]
Thus, the term following [tex]$-2$[/tex] is [tex]$-6$[/tex].
4. Finally, use [tex]$-6$[/tex] as the starting term and subtract 4 once more to find the third new term:
[tex]\[
-6 - 4 = -10
\][/tex]
Therefore, the term after [tex]$-6$[/tex] is [tex]$-10$[/tex].
Hence, the next three terms of the sequence [tex]$10, 6, 2, \ldots$[/tex] are:
[tex]\[
-2, -6, -10
\][/tex]
