To find the 8th term of the given arithmetic sequence [tex]\(5, \frac{11}{3}, \frac{7}{3}, \ldots\)[/tex], follow these steps:
1. Identify the first term [tex]\(a_1\)[/tex]:
[tex]\[ a_1 = 5 \][/tex]
2. Calculate the common difference [tex]\(d\)[/tex]:
To find the common difference, subtract the first term from the second term:
[tex]\[
d = \left( \frac{11}{3} \right) - 5
\][/tex]
Converting the first term to a fraction:
[tex]\[
5 = \frac{15}{3}
\][/tex]
Now, the common difference is:
[tex]\[
d = \frac{11}{3} - \frac{15}{3} = \frac{11 - 15}{3} = \frac{-4}{3} = -\frac{4}{3}
\][/tex]
3. Use the formula for the nth term of an arithmetic sequence:
The formula for the [tex]\(n\)[/tex]-th term [tex]\((a_n)\)[/tex] of an arithmetic sequence is:
[tex]\[
a_n = a_1 + (n - 1) \cdot d
\][/tex]
4. Substitute the known values into the formula to find the 8th term ([tex]\(a_8\)[/tex]):
[tex]\[
a_8 = 5 + (8 - 1) \cdot \left( -\frac{4}{3} \right)
\][/tex]
5. Simplify the expression:
Calculate [tex]\(8 - 1\)[/tex]:
[tex]\[
8 - 1 = 7
\][/tex]
Now, multiply 7 by the common difference:
[tex]\[
7 \cdot \left( -\frac{4}{3} \right) = -\frac{28}{3}
\][/tex]
Add this result to the first term:
[tex]\[
a_8 = 5 + \left( -\frac{28}{3} \right)
\][/tex]
6. Convert the first term to a fraction and combine:
Convert 5 to a fraction:
[tex]\[
5 = \frac{15}{3}
\][/tex]
Now add the two fractions:
[tex]\[
a_8 = \frac{15}{3} + \left( -\frac{28}{3} \right) = \frac{15 - 28}{3} = \frac{-13}{3}
\][/tex]
Therefore, the 8th term of the arithmetic sequence is:
[tex]\[
a_8 = -\frac{13}{3}
\][/tex]
Converting [tex]\(-\frac{13}{3}\)[/tex] to a decimal:
[tex]\[
a_8 = -4.333333333333334
\][/tex]
So, the 8th term is [tex]\(-\frac{13}{3}\)[/tex] or approximately [tex]\(-4.33\)[/tex].
