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].
