Answer :
Let's find the general terms [tex]\( t_n \)[/tex] of the given sequences one by one.
### (a) [tex]\( 4, 6, 8, 10, \ldots \)[/tex]
This is an arithmetic sequence with a common difference [tex]\( d = 6 - 4 = 2 \)[/tex].
The general term for an arithmetic sequence is given by:
[tex]\[ t_n = a + (n - 1) d \][/tex]
Here, [tex]\( a = 4 \)[/tex] and [tex]\( d = 2 \)[/tex].
So,
[tex]\[ t_n = 4 + (n - 1) \cdot 2 = 2n + 2 \][/tex]
### (b) [tex]\( 7, 11, 15, 19, 23, \ldots \)[/tex]
This is an arithmetic sequence with a common difference [tex]\( d = 11 - 7 = 4 \)[/tex].
Here, [tex]\( a = 7 \)[/tex] and [tex]\( d = 4 \)[/tex].
So,
[tex]\[ t_n = 7 + (n - 1) \cdot 4 = 4n + 3 \][/tex]
### (c) [tex]\( 2, 6, 10, 14, 18, \ldots \)[/tex]
This is an arithmetic sequence with a common difference [tex]\( d = 6 - 2 = 4 \)[/tex].
Here, [tex]\( a = 2 \)[/tex] and [tex]\( d = 4 \)[/tex].
So,
[tex]\[ t_n = 2 + (n - 1) \cdot 4 = 4n - 2 \][/tex]
### (d) [tex]\( 25, 22, 19, 16, \ldots \)[/tex]
This is an arithmetic sequence with a common difference [tex]\( d = 22 - 25 = -3 \)[/tex].
Here, [tex]\( a = 25 \)[/tex] and [tex]\( d = -3 \)[/tex].
So,
[tex]\[ t_n = 25 + (n - 1) \cdot (-3) = 28 - 3n \][/tex]
### (e) [tex]\( \frac{1}{3}, \frac{4}{5}, \frac{7}{7}, \frac{10}{9}, \ldots \)[/tex]
Let's express these ratios in terms of a general formula. The general term here is more complex.
The general term can be provided as:
[tex]\[ t_n = \frac{8406719304424927 n}{18014398509481984} - \frac{1200959900632133}{9007199254740992} \][/tex]
### (f) [tex]\( \frac{2}{7}, \frac{5}{8}, \frac{8}{9}, \frac{11}{10}, \ldots \)[/tex]
Similar to (e), this is another sequence with a fractional relationship.
The general term can be expressed as:
[tex]\[ t_n = \frac{3056014032858551 n}{9007199254740992} - \frac{241264265751991}{4503599627370496} \][/tex]
### (g) [tex]\( 40, 38, 36, 34, \ldots \)[/tex]
This is an arithmetic sequence with a common difference [tex]\( d = 38 - 40 = -2 \)[/tex].
Here, [tex]\( a = 40 \)[/tex] and [tex]\( d = -2 \)[/tex].
So,
[tex]\[ t_n = 40 + (n - 1) \cdot (-2) = 42 - 2n \][/tex]
### (h) [tex]\( \frac{2}{5}, \frac{4}{8}, \frac{6}{11}, \frac{8}{14}, \ldots \)[/tex]
Like the previous fractional sequences, this one also adheres to a specific ratio pattern.
The general term for this sequence is:
[tex]\[ t_n = \frac{900719925474099n}{9007199254740992} + \frac{1351079888211149}{4503599627370496} \][/tex]
These are the general terms for each of the given sequences.
### (a) [tex]\( 4, 6, 8, 10, \ldots \)[/tex]
This is an arithmetic sequence with a common difference [tex]\( d = 6 - 4 = 2 \)[/tex].
The general term for an arithmetic sequence is given by:
[tex]\[ t_n = a + (n - 1) d \][/tex]
Here, [tex]\( a = 4 \)[/tex] and [tex]\( d = 2 \)[/tex].
So,
[tex]\[ t_n = 4 + (n - 1) \cdot 2 = 2n + 2 \][/tex]
### (b) [tex]\( 7, 11, 15, 19, 23, \ldots \)[/tex]
This is an arithmetic sequence with a common difference [tex]\( d = 11 - 7 = 4 \)[/tex].
Here, [tex]\( a = 7 \)[/tex] and [tex]\( d = 4 \)[/tex].
So,
[tex]\[ t_n = 7 + (n - 1) \cdot 4 = 4n + 3 \][/tex]
### (c) [tex]\( 2, 6, 10, 14, 18, \ldots \)[/tex]
This is an arithmetic sequence with a common difference [tex]\( d = 6 - 2 = 4 \)[/tex].
Here, [tex]\( a = 2 \)[/tex] and [tex]\( d = 4 \)[/tex].
So,
[tex]\[ t_n = 2 + (n - 1) \cdot 4 = 4n - 2 \][/tex]
### (d) [tex]\( 25, 22, 19, 16, \ldots \)[/tex]
This is an arithmetic sequence with a common difference [tex]\( d = 22 - 25 = -3 \)[/tex].
Here, [tex]\( a = 25 \)[/tex] and [tex]\( d = -3 \)[/tex].
So,
[tex]\[ t_n = 25 + (n - 1) \cdot (-3) = 28 - 3n \][/tex]
### (e) [tex]\( \frac{1}{3}, \frac{4}{5}, \frac{7}{7}, \frac{10}{9}, \ldots \)[/tex]
Let's express these ratios in terms of a general formula. The general term here is more complex.
The general term can be provided as:
[tex]\[ t_n = \frac{8406719304424927 n}{18014398509481984} - \frac{1200959900632133}{9007199254740992} \][/tex]
### (f) [tex]\( \frac{2}{7}, \frac{5}{8}, \frac{8}{9}, \frac{11}{10}, \ldots \)[/tex]
Similar to (e), this is another sequence with a fractional relationship.
The general term can be expressed as:
[tex]\[ t_n = \frac{3056014032858551 n}{9007199254740992} - \frac{241264265751991}{4503599627370496} \][/tex]
### (g) [tex]\( 40, 38, 36, 34, \ldots \)[/tex]
This is an arithmetic sequence with a common difference [tex]\( d = 38 - 40 = -2 \)[/tex].
Here, [tex]\( a = 40 \)[/tex] and [tex]\( d = -2 \)[/tex].
So,
[tex]\[ t_n = 40 + (n - 1) \cdot (-2) = 42 - 2n \][/tex]
### (h) [tex]\( \frac{2}{5}, \frac{4}{8}, \frac{6}{11}, \frac{8}{14}, \ldots \)[/tex]
Like the previous fractional sequences, this one also adheres to a specific ratio pattern.
The general term for this sequence is:
[tex]\[ t_n = \frac{900719925474099n}{9007199254740992} + \frac{1351079888211149}{4503599627370496} \][/tex]
These are the general terms for each of the given sequences.
