To determine the common difference in the sequence [tex]\(9, 2.5, -4, -10.5, -17, \ldots\)[/tex], we need to look at the differences between successive terms.
Let's identify the terms of the sequence as follows:
- First term ([tex]\( T_1 \)[/tex]): [tex]\( 9 \)[/tex]
- Second term ([tex]\( T_2 \)[/tex]): [tex]\( 2.5 \)[/tex]
- Third term ([tex]\( T_3 \)[/tex]): [tex]\( -4 \)[/tex]
- Fourth term ([tex]\( T_4 \)[/tex]): [tex]\( -10.5 \)[/tex]
- Fifth term ([tex]\( T_5 \)[/tex]): [tex]\( -17 \)[/tex]
The common difference ([tex]\( d \)[/tex]) is obtained by subtracting a term from the term immediately preceding it. We can calculate it between the first two terms:
[tex]\[
d = T_2 - T_1
\][/tex]
Substituting the values from the sequence:
[tex]\[
d = 2.5 - 9
\][/tex]
Simplifying this calculation:
[tex]\[
d = -6.5
\][/tex]
Therefore, the common difference in the sequence is [tex]\( \boxed{-6.5} \)[/tex].
