To determine the value of [tex]\( x \)[/tex] such that [tex]\( x, 2, (x-10) \)[/tex] are the first three terms of an arithmetic sequence, we need to use the property of arithmetic sequences that the difference between consecutive terms is constant. This constant difference is referred to as the common difference.
Let's denote the first term by [tex]\( a_1 = x \)[/tex], the second term by [tex]\( a_2 = 2 \)[/tex], and the third term by [tex]\( a_3 = x - 10 \)[/tex].
The common difference [tex]\( d \)[/tex] can be found using the differences between consecutive terms:
[tex]\[
d = a_2 - a_1 = 2 - x
\][/tex]
[tex]\[
d = a_3 - a_2 = (x - 10) - 2 = x - 12
\][/tex]
Since the common difference must be the same, we set these two expressions for [tex]\( d \)[/tex] equal to each other:
[tex]\[
2 - x = x - 12
\][/tex]
Now we solve for [tex]\( x \)[/tex]:
1. Combine like terms to get all terms involving [tex]\( x \)[/tex] on one side of the equation:
[tex]\[
2 - x = x - 12
\][/tex]
[tex]\[
2 + 12 = x + x
\][/tex]
[tex]\[
14 = 2x
\][/tex]
2. Divide both sides of the equation by 2:
[tex]\[
7 = x
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{7} \)[/tex].
