To determine the common difference [tex]\( d \)[/tex] of the arithmetic sequence [tex]\( 1164.1, 1157, 1149.9, 1142.8 \)[/tex], we need to calculate the differences between consecutive terms and verify if they are consistent throughout the sequence.
1. First difference ([tex]\( d_1 \)[/tex]):
[tex]\[
d_1 = 1157 - 1164.1 = -7.1
\][/tex]
2. Second difference ([tex]\( d_2 \)[/tex]):
[tex]\[
d_2 = 1149.9 - 1157 = -7.1
\][/tex]
3. Third difference ([tex]\( d_3 \)[/tex]):
[tex]\[
d_3 = 1142.8 - 1149.9 = -7.1
\][/tex]
Examining the calculated differences:
- The first difference [tex]\( d_1 \)[/tex] is approximately [tex]\(-7.1\)[/tex].
- The second difference [tex]\( d_2 \)[/tex] is approximately [tex]\(-7.1\)[/tex].
- The third difference [tex]\( d_3 \)[/tex] is approximately [tex]\(-7.1\)[/tex].
However, since the differences are very close but not exactly the same ([tex]\(d_1 = -7.099999999999909\)[/tex], [tex]\(d_2 = -7.099999999999909\)[/tex], [tex]\(d_3 = -7.100000000000136\)[/tex]), we conclude that they are not perfectly consistent due to minute variations caused by rounding errors. Therefore, we cannot strictly assign a common difference to this sequence.
Given these observations, the correct answer would be that the sequence has no common difference.
Answer:
d. The sequence has no common difference.
