Answer :
Let's address each part of the question step-by-step.
### Part 1: Finding Two More Terms
#### (a) Sequence: [tex]\(1, 2, 3, 4, \ldots\)[/tex]
The given sequence increases by 1 each time.
- Next two terms: [tex]\(5, 6\)[/tex]
So, the sequence becomes: [tex]\(1, 2, 3, 4, 5, 6\)[/tex]
#### (b) Sequence: [tex]\(5, 10, 15, 20, 25, \ldots\)[/tex]
The given sequence increases by 5 each time.
- Next two terms: [tex]\(30, 35\)[/tex]
So, the sequence becomes: [tex]\(5, 10, 15, 20, 25, 30, 35\)[/tex]
#### (c) Sequence: [tex]\(2, 4, 6, 8, 10, \ldots\)[/tex]
The given sequence increases by 2 each time.
- Next two terms: [tex]\(12, 14\)[/tex]
So, the sequence becomes: [tex]\(2, 4, 6, 8, 10, 12, 14\)[/tex]
#### (d) Sequence: [tex]\(22, 20, 18, 16, 14, \ldots\)[/tex]
The given sequence decreases by 2 each time.
- Next two terms: [tex]\(12, 10\)[/tex]
So, the sequence becomes: [tex]\(22, 20, 18, 16, 14, 12, 10\)[/tex]
#### (e) Sequence: [tex]\(1, 2, 4, 8, 16, \ldots\)[/tex]
The given sequence is multiplied by 2 each time.
- Next two terms: [tex]\(32, 64\)[/tex]
So, the sequence becomes: [tex]\(1, 2, 4, 8, 16, 32, 64\)[/tex]
#### (f) Sequence: [tex]\(2, 5, 9, 14, 20, \ldots\)[/tex]
The given sequence increases by increments of 3, 4, 5, ...
- Next two terms: [tex]\(27, 35\)[/tex]
So, the sequence becomes: [tex]\(2, 5, 9, 14, 20, 27, 35\)[/tex]
#### (g) Sequence: [tex]\(-8, -6, -4, \ldots\)[/tex]
The given sequence increases by 2 each time.
- Next two terms: [tex]\(-2, 0\)[/tex]
So, the sequence becomes: [tex]\(-8, -6, -4, -2, 0\)[/tex]
#### (h) Sequence: [tex]\(\frac{1}{3}, \frac{4}{5}, \frac{7}{7}, \frac{10}{9}, \ldots\)[/tex]
Identifying the terms does not appear straightforward, but using the provided result:
- Next two terms: [tex]\(\frac{13}{11}, \frac{16}{13}\)[/tex]
So, the sequence becomes: [tex]\(\frac{1}{3}, \frac{4}{5}, \frac{7}{7}, \frac{10}{9}, \frac{13}{11}, \frac{16}{13}\)[/tex]
### Part 2: General Terms [tex]\( t_n \)[/tex]
Let's determine the general term for each sequence.
#### (a) Sequence: [tex]\(4, 6, 8, 10, \ldots\)[/tex]
This is an arithmetic sequence with a common difference of 2 and a first term of 4.
[tex]\[ t_n = 2n + 2 \][/tex]
#### (b) Sequence: [tex]\(7, 11, 15, 19, 23, \ldots\)[/tex]
This is an arithmetic sequence with a common difference of 4 and a first term of 7.
[tex]\[ t_n = 4n + 3 \][/tex]
#### (c) Sequence: [tex]\(2, 6, 10, 14, 18, \ldots\)[/tex]
This is an arithmetic sequence with a common difference of 4 and a first term of 2.
[tex]\[ t_n = 4n - 2 \][/tex]
#### (d) Sequence: [tex]\(25, 22, 19, 16, \ldots\)[/tex]
This is an arithmetic sequence with a common difference of -3 and a first term of 25.
[tex]\[ t_n = 28 - 3n \][/tex]
#### (e) Sequence: [tex]\(\frac{1}{3}, \frac{4}{5}, \frac{7}{7}, \frac{10}{9}, \ldots\)[/tex]
The general term cannot be determined as a simple arithmetic or geometric sequence.
[tex]\[ t_n = \text{Cannot be determined as a simple arithmetic or geometric sequence} \][/tex]
#### (f) Sequence: [tex]\(\frac{2}{7}, \frac{5}{8}, \frac{8}{9}, \frac{11}{10}, \ldots\)[/tex]
The general term cannot be determined as a simple arithmetic or geometric sequence.
[tex]\[ t_n = \text{Cannot be determined as a simple arithmetic or geometric sequence} \][/tex]
#### (g) Sequence: [tex]\(40, 38, 36, 34, \ldots\)[/tex]
This is an arithmetic sequence with a common difference of -2 and a first term of 40.
[tex]\[ t_n = 42 - 2n \][/tex]
#### (h) Sequence: [tex]\(\frac{2}{5}, \frac{4}{8}, \frac{6}{11}, \frac{8}{14}, \ldots\)[/tex]
The general term cannot be determined as a simple arithmetic or geometric sequence.
[tex]\[ t_n = \text{Cannot be determined as a simple arithmetic or geometric sequence} \][/tex]
This concludes our detailed solution.
### Part 1: Finding Two More Terms
#### (a) Sequence: [tex]\(1, 2, 3, 4, \ldots\)[/tex]
The given sequence increases by 1 each time.
- Next two terms: [tex]\(5, 6\)[/tex]
So, the sequence becomes: [tex]\(1, 2, 3, 4, 5, 6\)[/tex]
#### (b) Sequence: [tex]\(5, 10, 15, 20, 25, \ldots\)[/tex]
The given sequence increases by 5 each time.
- Next two terms: [tex]\(30, 35\)[/tex]
So, the sequence becomes: [tex]\(5, 10, 15, 20, 25, 30, 35\)[/tex]
#### (c) Sequence: [tex]\(2, 4, 6, 8, 10, \ldots\)[/tex]
The given sequence increases by 2 each time.
- Next two terms: [tex]\(12, 14\)[/tex]
So, the sequence becomes: [tex]\(2, 4, 6, 8, 10, 12, 14\)[/tex]
#### (d) Sequence: [tex]\(22, 20, 18, 16, 14, \ldots\)[/tex]
The given sequence decreases by 2 each time.
- Next two terms: [tex]\(12, 10\)[/tex]
So, the sequence becomes: [tex]\(22, 20, 18, 16, 14, 12, 10\)[/tex]
#### (e) Sequence: [tex]\(1, 2, 4, 8, 16, \ldots\)[/tex]
The given sequence is multiplied by 2 each time.
- Next two terms: [tex]\(32, 64\)[/tex]
So, the sequence becomes: [tex]\(1, 2, 4, 8, 16, 32, 64\)[/tex]
#### (f) Sequence: [tex]\(2, 5, 9, 14, 20, \ldots\)[/tex]
The given sequence increases by increments of 3, 4, 5, ...
- Next two terms: [tex]\(27, 35\)[/tex]
So, the sequence becomes: [tex]\(2, 5, 9, 14, 20, 27, 35\)[/tex]
#### (g) Sequence: [tex]\(-8, -6, -4, \ldots\)[/tex]
The given sequence increases by 2 each time.
- Next two terms: [tex]\(-2, 0\)[/tex]
So, the sequence becomes: [tex]\(-8, -6, -4, -2, 0\)[/tex]
#### (h) Sequence: [tex]\(\frac{1}{3}, \frac{4}{5}, \frac{7}{7}, \frac{10}{9}, \ldots\)[/tex]
Identifying the terms does not appear straightforward, but using the provided result:
- Next two terms: [tex]\(\frac{13}{11}, \frac{16}{13}\)[/tex]
So, the sequence becomes: [tex]\(\frac{1}{3}, \frac{4}{5}, \frac{7}{7}, \frac{10}{9}, \frac{13}{11}, \frac{16}{13}\)[/tex]
### Part 2: General Terms [tex]\( t_n \)[/tex]
Let's determine the general term for each sequence.
#### (a) Sequence: [tex]\(4, 6, 8, 10, \ldots\)[/tex]
This is an arithmetic sequence with a common difference of 2 and a first term of 4.
[tex]\[ t_n = 2n + 2 \][/tex]
#### (b) Sequence: [tex]\(7, 11, 15, 19, 23, \ldots\)[/tex]
This is an arithmetic sequence with a common difference of 4 and a first term of 7.
[tex]\[ t_n = 4n + 3 \][/tex]
#### (c) Sequence: [tex]\(2, 6, 10, 14, 18, \ldots\)[/tex]
This is an arithmetic sequence with a common difference of 4 and a first term of 2.
[tex]\[ t_n = 4n - 2 \][/tex]
#### (d) Sequence: [tex]\(25, 22, 19, 16, \ldots\)[/tex]
This is an arithmetic sequence with a common difference of -3 and a first term of 25.
[tex]\[ t_n = 28 - 3n \][/tex]
#### (e) Sequence: [tex]\(\frac{1}{3}, \frac{4}{5}, \frac{7}{7}, \frac{10}{9}, \ldots\)[/tex]
The general term cannot be determined as a simple arithmetic or geometric sequence.
[tex]\[ t_n = \text{Cannot be determined as a simple arithmetic or geometric sequence} \][/tex]
#### (f) Sequence: [tex]\(\frac{2}{7}, \frac{5}{8}, \frac{8}{9}, \frac{11}{10}, \ldots\)[/tex]
The general term cannot be determined as a simple arithmetic or geometric sequence.
[tex]\[ t_n = \text{Cannot be determined as a simple arithmetic or geometric sequence} \][/tex]
#### (g) Sequence: [tex]\(40, 38, 36, 34, \ldots\)[/tex]
This is an arithmetic sequence with a common difference of -2 and a first term of 40.
[tex]\[ t_n = 42 - 2n \][/tex]
#### (h) Sequence: [tex]\(\frac{2}{5}, \frac{4}{8}, \frac{6}{11}, \frac{8}{14}, \ldots\)[/tex]
The general term cannot be determined as a simple arithmetic or geometric sequence.
[tex]\[ t_n = \text{Cannot be determined as a simple arithmetic or geometric sequence} \][/tex]
This concludes our detailed solution.
