Westonci.ca is the best place to get answers to your questions, provided by a community of experienced and knowledgeable experts. Discover reliable solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
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.
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.