To determine which of the given sequences is an arithmetic sequence, we need to check if the differences between consecutive terms are consistent.
An arithmetic sequence is one in which the difference between any two consecutive terms is constant. This difference is known as the common difference.
Let's analyze each sequence step-by-step:
1. Sequence: [tex]\(\{724, -362, 181, -90.5\}\)[/tex]
- Difference between the first and second terms: [tex]\(-362 - 724 = -1086\)[/tex]
- Difference between the second and third terms: [tex]\(181 - (-362) = 181 + 362 = 543\)[/tex]
- Difference between the third and fourth terms: [tex]\(-90.5 - 181 = -271.5\)[/tex]
Since the differences are [tex]\(-1086\)[/tex], [tex]\(543\)[/tex], and [tex]\(-271.5\)[/tex] which are not consistent, this sequence is not arithmetic.
2. Sequence: [tex]\(\{232, 354, 476, 598, \ldots\}\)[/tex]
- Difference between the first and second terms: [tex]\(354 - 232 = 122\)[/tex]
- Difference between the second and third terms: [tex]\(476 - 354 = 122\)[/tex]
- Difference between the third and fourth terms: [tex]\(598 - 476 = 122\)[/tex]
Since the differences are [tex]\(122\)[/tex], [tex]\(122\)[/tex], and [tex]\(122\)[/tex] which are consistent, this sequence is arithmetic with a common difference of [tex]\(122\)[/tex].
3. Sequence: [tex]\(\{691, 632, 573, 514\}\)[/tex]
- Difference between the first and second terms: [tex]\(632 - 691 = -59\)[/tex]
- Difference between the second and third terms: [tex]\(573 - 632 = -59\)[/tex]
- Difference between the third and fourth terms: [tex]\(514 - 573 = -59\)[/tex]
Since the differences are [tex]\(-59\)[/tex], [tex]\(-59\)[/tex], and [tex]\(-59\)[/tex] which are consistent, this sequence is arithmetic with a common difference of [tex]\(-59\)[/tex].
4. Sequence: [tex]\(\{845, 169, 33.8, 6.76, \ldots\}\)[/tex]
- Difference between the first and second terms: [tex]\(169 - 845 = -676\)[/tex]
- Difference between the second and third terms: [tex]\(33.8 - 169 = -135.2\)[/tex]
- Difference between the third and fourth terms: [tex]\(6.76 - 33.8 = -27.04\)[/tex]
Since the differences are [tex]\(-676\)[/tex], [tex]\(-135.2\)[/tex], and [tex]\(-27.04\)[/tex] which are not consistent, this sequence is not arithmetic.
Out of all the sequences provided, the second sequence [tex]\(\{232, 354, 476, 598, \ldots\}\)[/tex] is a consistent arithmetic sequence with a common difference of [tex]\(122\)[/tex].
Thus, the infinite arithmetic sequence among the given options is:
[tex]$\{232, 354, 476, 598, \ldots\}$[/tex]
