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Find the 205th term of the sequence.

Given:
- Common difference: [tex]12 - 8 = 4[/tex]
- Sequence: [tex]-8, -4, 0, 4, 8, 12, \ldots[/tex]

Use the formula for the nth term of an arithmetic sequence:
[tex]\[a_n = a + (n - 1)d\][/tex]

Where:
- [tex]a[/tex] is the first term ([tex]\(-8\)[/tex])
- [tex]d[/tex] is the common difference ([tex]\(4\)[/tex])
- [tex]n[/tex] is the desired term (205)

Plugging in the values:
[tex]\[a_{205} = -8 + (205 - 1) \times 4\][/tex]

Calculate the 205th term.



Answer :

GinnyAnswer
To find the desired 205th term of an arithmetic sequence, we can identify the following information from the problem:

1. First term ([tex]\(a\)[/tex]): The first term of the sequence is given as [tex]\( -8 \)[/tex].
2. Common difference ([tex]\(d\)[/tex]): The common difference can be calculated as [tex]\( 12 - 8 = 4 \)[/tex].
3. Desired term number ([tex]\(n\)[/tex]): We are looking for the 205th term, so [tex]\( n = 205 \)[/tex].

The formula to find the [tex]\( n \)[/tex]-th term ([tex]\( a_n \)[/tex]) in an arithmetic sequence is:
[tex]\[ a_n = a + (n - 1) \cdot d \][/tex]

Here’s a step-by-step application of this formula:

1. Identify the first term ([tex]\(a\)[/tex]):
[tex]\[ a = -8 \][/tex]

2. Identify the common difference ([tex]\(d\)[/tex]):
[tex]\[ d = 4 \][/tex]

3. Identify the term number ([tex]\(n\)[/tex]):
[tex]\[ n = 205 \][/tex]

4. Substitute these values into the formula:
[tex]\[ a_n = -8 + (205 - 1) \cdot 4 \][/tex]

5. Simplify inside the parentheses:
[tex]\[ a_n = -8 + 204 \cdot 4 \][/tex]

6. Multiply:
[tex]\[ 204 \cdot 4 = 816 \][/tex]

7. Add:
[tex]\[ a_n = -8 + 816 = 808 \][/tex]

Thus, the 205th term in the arithmetic sequence is [tex]\( \boxed{808} \)[/tex].

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