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Answer :
To find the 50th term of the arithmetic sequence: 6, 2, -2, ..., follow these steps:
1. Identify the first term ([tex]\(a_1\)[/tex]):
In the sequence, the first term is [tex]\(a_1 = 6\)[/tex].
2. Determine the common difference ([tex]\(d\)[/tex]):
The common difference in an arithmetic sequence is the difference between consecutive terms.
[tex]\[
d = 2 - 6 = -4
\][/tex]
3. Use the formula for the nth term of an arithmetic sequence:
The formula is given by:
[tex]\[
a_n = a_1 + (n-1) \cdot d
\][/tex]
where [tex]\(a_n\)[/tex] is the nth term, [tex]\(a_1\)[/tex] is the first term, [tex]\(n\)[/tex] is the term number, and [tex]\(d\)[/tex] is the common difference.
4. Plug the values into the formula to find the 50th term ([tex]\(a_{50}\)[/tex]):
[tex]\[
a_{50} = 6 + (50 - 1) \cdot (-4)
\][/tex]
[tex]\[
a_{50} = 6 + 49 \cdot (-4)
\][/tex]
[tex]\[
a_{50} = 6 - 196
\][/tex]
[tex]\[
a_{50} = -190
\][/tex]
So, the 50th term of the arithmetic sequence is [tex]\(-190\)[/tex].
1. Identify the first term ([tex]\(a_1\)[/tex]):
In the sequence, the first term is [tex]\(a_1 = 6\)[/tex].
2. Determine the common difference ([tex]\(d\)[/tex]):
The common difference in an arithmetic sequence is the difference between consecutive terms.
[tex]\[
d = 2 - 6 = -4
\][/tex]
3. Use the formula for the nth term of an arithmetic sequence:
The formula is given by:
[tex]\[
a_n = a_1 + (n-1) \cdot d
\][/tex]
where [tex]\(a_n\)[/tex] is the nth term, [tex]\(a_1\)[/tex] is the first term, [tex]\(n\)[/tex] is the term number, and [tex]\(d\)[/tex] is the common difference.
4. Plug the values into the formula to find the 50th term ([tex]\(a_{50}\)[/tex]):
[tex]\[
a_{50} = 6 + (50 - 1) \cdot (-4)
\][/tex]
[tex]\[
a_{50} = 6 + 49 \cdot (-4)
\][/tex]
[tex]\[
a_{50} = 6 - 196
\][/tex]
[tex]\[
a_{50} = -190
\][/tex]
So, the 50th term of the arithmetic sequence is [tex]\(-190\)[/tex].
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