Answer :
To find the first five terms of each sequence defined by the given general terms with [tex]\( n \)[/tex] representing the natural number, we proceed as follows:
### (a) Sequence: [tex]\( t_n = 2n + 4 \)[/tex]
Calculate the first five terms for [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = 2(1) + 4 = 6 \)[/tex]
2. [tex]\( t_2 = 2(2) + 4 = 8 \)[/tex]
3. [tex]\( t_3 = 2(3) + 4 = 10 \)[/tex]
4. [tex]\( t_4 = 2(4) + 4 = 12 \)[/tex]
5. [tex]\( t_5 = 2(5) + 4 = 14 \)[/tex]
So, the sequence is: 6, 8, 10, 12, 14
### (b) Sequence: [tex]\( t_n = 3n - 1 \)[/tex]
Calculate the first five terms for [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = 3(1) - 1 = 2 \)[/tex]
2. [tex]\( t_2 = 3(2) - 1 = 5 \)[/tex]
3. [tex]\( t_3 = 3(3) - 1 = 8 \)[/tex]
4. [tex]\( t_4 = 3(4) - 1 = 11 \)[/tex]
5. [tex]\( t_5 = 3(5) - 1 = 14 \)[/tex]
So, the sequence is: 2, 5, 8, 11, 14
### (c) Sequence: [tex]\( t_n = 3^n \)[/tex]
Calculate the first five terms for [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = 3^1 = 3 \)[/tex]
2. [tex]\( t_2 = 3^2 = 9 \)[/tex]
3. [tex]\( t_3 = 3^3 = 27 \)[/tex]
4. [tex]\( t_4 = 3^4 = 81 \)[/tex]
5. [tex]\( t_5 = 3^5 = 243 \)[/tex]
So, the sequence is: 3, 9, 27, 81, 243
### (d) Sequence: [tex]\( t_n = n^2 - 1 \)[/tex]
Calculate the first five terms for [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = 1^2 - 1 = 0 \)[/tex]
2. [tex]\( t_2 = 2^2 - 1 = 3 \)[/tex]
3. [tex]\( t_3 = 3^2 - 1 = 8 \)[/tex]
4. [tex]\( t_4 = 4^2 - 1 = 15 \)[/tex]
5. [tex]\( t_5 = 5^2 - 1 = 24 \)[/tex]
So, the sequence is: 0, 3, 8, 15, 24
### (e) Sequence: [tex]\( t_n = (-1)^n \cdot n^2 \)[/tex]
Calculate the first five terms for [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = (-1)^1 \cdot 1^2 = -1 \)[/tex]
2. [tex]\( t_2 = (-1)^2 \cdot 2^2 = 4 \)[/tex]
3. [tex]\( t_3 = (-1)^3 \cdot 3^2 = -9 \)[/tex]
4. [tex]\( t_4 = (-1)^4 \cdot 4^2 = 16 \)[/tex]
5. [tex]\( t_5 = (-1)^5 \cdot 5^2 = -25 \)[/tex]
So, the sequence is: -1, 4, -9, 16, -25
### (f) Sequence: [tex]\( t_n = n^2 + 2n + 3 \)[/tex]
Calculate the first five terms for [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = 1^2 + 2(1) + 3 = 6 \)[/tex]
2. [tex]\( t_2 = 2^2 + 2(2) + 3 = 11 \)[/tex]
3. [tex]\( t_3 = 3^2 + 2(3) + 3 = 18 \)[/tex]
4. [tex]\( t_4 = 4^2 + 2(4) + 3 = 27 \)[/tex]
5. [tex]\( t_5 = 5^2 + 2(5) + 3 = 38 \)[/tex]
So, the sequence is: 6, 11, 18, 27, 38
### (g) Sequence: [tex]\( t_n = 3n^2 - 5 \)[/tex]
Calculate the first five terms for [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = 3(1)^2 - 5 = -2 \)[/tex]
2. [tex]\( t_2 = 3(2)^2 - 5 = 7 \)[/tex]
3. [tex]\( t_3 = 3(3)^2 - 5 = 22 \)[/tex]
4. [tex]\( t_4 = 3(4)^2 - 5 = 43 \)[/tex]
5. [tex]\( t_5 = 3(5)^2 - 5 = 70 \)[/tex]
So, the sequence is: -2, 7, 22, 43, 70
In summary, the first five terms for each of the sequences are:
1. 6, 8, 10, 12, 14
2. 2, 5, 8, 11, 14
3. 3, 9, 27, 81, 243
4. 0, 3, 8, 15, 24
5. -1, 4, -9, 16, -25
6. 6, 11, 18, 27, 38
7. -2, 7, 22, 43, 70
### (a) Sequence: [tex]\( t_n = 2n + 4 \)[/tex]
Calculate the first five terms for [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = 2(1) + 4 = 6 \)[/tex]
2. [tex]\( t_2 = 2(2) + 4 = 8 \)[/tex]
3. [tex]\( t_3 = 2(3) + 4 = 10 \)[/tex]
4. [tex]\( t_4 = 2(4) + 4 = 12 \)[/tex]
5. [tex]\( t_5 = 2(5) + 4 = 14 \)[/tex]
So, the sequence is: 6, 8, 10, 12, 14
### (b) Sequence: [tex]\( t_n = 3n - 1 \)[/tex]
Calculate the first five terms for [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = 3(1) - 1 = 2 \)[/tex]
2. [tex]\( t_2 = 3(2) - 1 = 5 \)[/tex]
3. [tex]\( t_3 = 3(3) - 1 = 8 \)[/tex]
4. [tex]\( t_4 = 3(4) - 1 = 11 \)[/tex]
5. [tex]\( t_5 = 3(5) - 1 = 14 \)[/tex]
So, the sequence is: 2, 5, 8, 11, 14
### (c) Sequence: [tex]\( t_n = 3^n \)[/tex]
Calculate the first five terms for [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = 3^1 = 3 \)[/tex]
2. [tex]\( t_2 = 3^2 = 9 \)[/tex]
3. [tex]\( t_3 = 3^3 = 27 \)[/tex]
4. [tex]\( t_4 = 3^4 = 81 \)[/tex]
5. [tex]\( t_5 = 3^5 = 243 \)[/tex]
So, the sequence is: 3, 9, 27, 81, 243
### (d) Sequence: [tex]\( t_n = n^2 - 1 \)[/tex]
Calculate the first five terms for [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = 1^2 - 1 = 0 \)[/tex]
2. [tex]\( t_2 = 2^2 - 1 = 3 \)[/tex]
3. [tex]\( t_3 = 3^2 - 1 = 8 \)[/tex]
4. [tex]\( t_4 = 4^2 - 1 = 15 \)[/tex]
5. [tex]\( t_5 = 5^2 - 1 = 24 \)[/tex]
So, the sequence is: 0, 3, 8, 15, 24
### (e) Sequence: [tex]\( t_n = (-1)^n \cdot n^2 \)[/tex]
Calculate the first five terms for [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = (-1)^1 \cdot 1^2 = -1 \)[/tex]
2. [tex]\( t_2 = (-1)^2 \cdot 2^2 = 4 \)[/tex]
3. [tex]\( t_3 = (-1)^3 \cdot 3^2 = -9 \)[/tex]
4. [tex]\( t_4 = (-1)^4 \cdot 4^2 = 16 \)[/tex]
5. [tex]\( t_5 = (-1)^5 \cdot 5^2 = -25 \)[/tex]
So, the sequence is: -1, 4, -9, 16, -25
### (f) Sequence: [tex]\( t_n = n^2 + 2n + 3 \)[/tex]
Calculate the first five terms for [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = 1^2 + 2(1) + 3 = 6 \)[/tex]
2. [tex]\( t_2 = 2^2 + 2(2) + 3 = 11 \)[/tex]
3. [tex]\( t_3 = 3^2 + 2(3) + 3 = 18 \)[/tex]
4. [tex]\( t_4 = 4^2 + 2(4) + 3 = 27 \)[/tex]
5. [tex]\( t_5 = 5^2 + 2(5) + 3 = 38 \)[/tex]
So, the sequence is: 6, 11, 18, 27, 38
### (g) Sequence: [tex]\( t_n = 3n^2 - 5 \)[/tex]
Calculate the first five terms for [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
1. [tex]\( t_1 = 3(1)^2 - 5 = -2 \)[/tex]
2. [tex]\( t_2 = 3(2)^2 - 5 = 7 \)[/tex]
3. [tex]\( t_3 = 3(3)^2 - 5 = 22 \)[/tex]
4. [tex]\( t_4 = 3(4)^2 - 5 = 43 \)[/tex]
5. [tex]\( t_5 = 3(5)^2 - 5 = 70 \)[/tex]
So, the sequence is: -2, 7, 22, 43, 70
In summary, the first five terms for each of the sequences are:
1. 6, 8, 10, 12, 14
2. 2, 5, 8, 11, 14
3. 3, 9, 27, 81, 243
4. 0, 3, 8, 15, 24
5. -1, 4, -9, 16, -25
6. 6, 11, 18, 27, 38
7. -2, 7, 22, 43, 70
