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Answer :
To calculate the 42nd term of the given arithmetic sequence, we need to identify the first term and the common difference, and then use the formula for the nth term of an arithmetic sequence.
1. Identify the first term ([tex]\(a_1\)[/tex]):
The first term of the sequence is given as [tex]\(4.5\)[/tex].
2. Find the common difference ([tex]\(d\)[/tex]):
To find the common difference, subtract the first term from the second term:
[tex]\[
d = 2 - 4.5 = -2.5
\][/tex]
3. Use the formula for the nth term of an arithmetic sequence:
The formula to find the nth term ([tex]\(a_n\)[/tex]) is:
[tex]\[
a_n = a_1 + (n-1) \cdot d
\][/tex]
Here, we want to find the 42nd term ([tex]\(a_{42}\)[/tex]), so substitute [tex]\(n = 42\)[/tex], [tex]\(a_1 = 4.5\)[/tex], and [tex]\(d = -2.5\)[/tex] into the formula:
[tex]\[
a_{42} = 4.5 + (42 - 1) \cdot (-2.5)
\][/tex]
[tex]\[
a_{42} = 4.5 + 41 \cdot (-2.5)
\][/tex]
[tex]\[
a_{42} = 4.5 - 102.5
\][/tex]
[tex]\[
a_{42} = -98
\][/tex]
Therefore, the 42nd term of the sequence is [tex]\(-98\)[/tex].
1. Identify the first term ([tex]\(a_1\)[/tex]):
The first term of the sequence is given as [tex]\(4.5\)[/tex].
2. Find the common difference ([tex]\(d\)[/tex]):
To find the common difference, subtract the first term from the second term:
[tex]\[
d = 2 - 4.5 = -2.5
\][/tex]
3. Use the formula for the nth term of an arithmetic sequence:
The formula to find the nth term ([tex]\(a_n\)[/tex]) is:
[tex]\[
a_n = a_1 + (n-1) \cdot d
\][/tex]
Here, we want to find the 42nd term ([tex]\(a_{42}\)[/tex]), so substitute [tex]\(n = 42\)[/tex], [tex]\(a_1 = 4.5\)[/tex], and [tex]\(d = -2.5\)[/tex] into the formula:
[tex]\[
a_{42} = 4.5 + (42 - 1) \cdot (-2.5)
\][/tex]
[tex]\[
a_{42} = 4.5 + 41 \cdot (-2.5)
\][/tex]
[tex]\[
a_{42} = 4.5 - 102.5
\][/tex]
[tex]\[
a_{42} = -98
\][/tex]
Therefore, the 42nd term of the sequence is [tex]\(-98\)[/tex].
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