To determine the common difference in an arithmetic progression (AP), we need to look at the difference between successive terms. The common difference [tex]\(d\)[/tex] is defined as the difference between any two consecutive terms in the sequence.
Consider the given AP: [tex]\(x-1, x-2, x-3, x-4, \ldots\)[/tex].
To find the common difference, we subtract the second term from the first term:
- First term: [tex]\(x-1\)[/tex]
- Second term: [tex]\(x-2\)[/tex]
The common difference [tex]\(d\)[/tex] is:
[tex]\[ d = (x-2) - (x-1) \][/tex]
Now, let's simplify the expression:
[tex]\[ d = x - 2 - (x - 1) \][/tex]
[tex]\[ d = x - 2 - x + 1 \][/tex]
[tex]\[ d = x - x - 2 + 1 \][/tex]
[tex]\[ d = -1 \][/tex]
Therefore, the common difference [tex]\(d\)[/tex] of the arithmetic progression [tex]\(x-1, x-2, x-3, x-4, \ldots\)[/tex] is [tex]\(-1\)[/tex].
So, the correct answer is:
c) -1
