Let's find the [tex]\(n\)[/tex]th term for each of the given sequences:
a. [tex]\(3, 6, 9, 12, \ldots\)[/tex]
This is an arithmetic sequence where the first term [tex]\(a_1 = 3\)[/tex] and the common difference [tex]\(d = 3\)[/tex].
The general formula for the [tex]\(n\)[/tex]th term of an arithmetic sequence is given by:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
Substituting the known values:
[tex]\[ a_n = 3 + (n - 1) \cdot 3 \][/tex]
Thus, the [tex]\(10\)[/tex]th term is:
[tex]\[ a_{10} = 3 + (10 - 1) \cdot 3 = 3 + 9 \cdot 3 = 3 + 27 = 30 \][/tex]
b. [tex]\(25, 22, 19, 16, \ldots\)[/tex]
This is an arithmetic sequence where the first term [tex]\(a_1 = 25\)[/tex] and the common difference [tex]\(d = -3\)[/tex].
Using the general formula:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
Substituting the known values:
[tex]\[ a_n = 25 + (n - 1) \cdot (-3) \][/tex]
Thus, the [tex]\(10\)[/tex]th term is:
[tex]\[ a_{10} = 25 + (10 - 1) \cdot (-3) = 25 + 9 \cdot (-3) = 25 - 27 = -2 \][/tex]
c. [tex]\(1, 4, 9, 16, \ldots\)[/tex]
This appears to be a quadratic sequence where the [tex]\(n\)[/tex]th term is:
[tex]\[ a_n = n^2 \][/tex]
Thus, the [tex]\(10\)[/tex]th term is:
[tex]\[ a_{10} = 10^2 = 100 \][/tex]
d. [tex]\(\frac{1}{3}, \frac{4}{5}, \frac{7}{7}, \frac{10}{9}, \ldots\)[/tex]
The [tex]\(n\)[/tex]th term of this sequence can be noted as:
[tex]\[ a_n = \frac{3n - 2}{2n + 1} \][/tex]
Thus, the [tex]\(10\)[/tex]th term is:
[tex]\[ a_{10} = \frac{3 \cdot 10 - 2}{2 \cdot 10 + 1} = \frac{30 - 2}{20 + 1} = \frac{28}{21} = \frac{4}{3} \approx 1.3333333333333333 \][/tex]
e. [tex]\(5, 2, -1, -4, \ldots\)[/tex]
This is another arithmetic sequence where the first term [tex]\(a_1 = 5\)[/tex] and the common difference [tex]\(d = -3\)[/tex].
Using the general formula:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
Substituting the known values:
[tex]\[ a_n = 5 + (n - 1) \cdot (-3) \][/tex]
Thus, the [tex]\(10\)[/tex]th term is:
[tex]\[ a_{10} = 5 + (10 - 1) \cdot (-3) = 5 + 9 \cdot (-3) = 5 - 27 = -22 \][/tex]
f. [tex]\(6, 30, 150, 750, \ldots\)[/tex]
This is a geometric sequence where the first term [tex]\(a_1 = 6\)[/tex] and the common ratio [tex]\(r = 5\)[/tex].
The general formula for the [tex]\(n\)[/tex]th term of a geometric sequence is given by:
[tex]\[ a_n = a_1 \cdot r^{(n - 1)} \][/tex]
Substituting the known values:
[tex]\[ a_n = 6 \cdot 5^{(n - 1)} \][/tex]
Thus, the [tex]\(10\)[/tex]th term is:
[tex]\[ a_{10} = 6 \cdot 5^{(10 - 1)} = 6 \cdot 5^9 = 6 \cdot 1953125 = 11718750 \][/tex]
Therefore, the [tex]\(10\)[/tex]th terms of the sequences are:
a. [tex]\(30\)[/tex]
b. [tex]\(-2\)[/tex]
c. [tex]\(100\)[/tex]
d. [tex]\(1.3333333333333333\)[/tex]
e. [tex]\(-22\)[/tex]
f. [tex]\(11718750\)[/tex]
