Answer :
Sure! Let's find the first 5 terms for each of the given sequences based on the provided [tex]\( n \)[/tex]th term formulas.
1. For the sequence [tex]\( a_n = n + 4 \)[/tex]:
- For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = 1 + 4 = 5 \][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = 2 + 4 = 6 \][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = 3 + 4 = 7 \][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = 4 + 4 = 8 \][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = 5 + 4 = 9 \][/tex]
So, the first 5 terms are: [tex]\( [5, 6, 7, 8, 9] \)[/tex].
2. For the sequence [tex]\( a_n = 2n - 1 \)[/tex]:
- For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = 2 \cdot 1 - 1 = 1 \][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = 2 \cdot 2 - 1 = 3 \][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = 2 \cdot 3 - 1 = 5 \][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = 2 \cdot 4 - 1 = 7 \][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = 2 \cdot 5 - 1 = 9 \][/tex]
So, the first 5 terms are: [tex]\( [1, 3, 5, 7, 9] \)[/tex].
3. For the sequence [tex]\( a_n = 12 - 3n \)[/tex]:
- For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = 12 - 3 \cdot 1 = 9 \][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = 12 - 3 \cdot 2 = 6 \][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = 12 - 3 \cdot 3 = 3 \][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = 12 - 3 \cdot 4 = 0 \][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = 12 - 3 \cdot 5 = -3 \][/tex]
So, the first 5 terms are: [tex]\( [9, 6, 3, 0, -3] \)[/tex].
4. For the sequence [tex]\( a_n = 3^n \)[/tex]:
- For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = 3^1 = 3 \][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = 3^2 = 9 \][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = 3^3 = 27 \][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = 3^4 = 81 \][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = 3^5 = 243 \][/tex]
So, the first 5 terms are: [tex]\( [3, 9, 27, 81, 243] \)[/tex].
5. For the sequence [tex]\( a_n = (-2)^n \)[/tex]:
- For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = (-2)^1 = -2 \][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = (-2)^2 = 4 \][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = (-2)^3 = -8 \][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = (-2)^4 = 16 \][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = (-2)^5 = -32 \][/tex]
So, the first 5 terms are: [tex]\( [-2, 4, -8, 16, -32] \)[/tex].
In summary, the first 5 terms for each sequence are:
1. [tex]\( a_n = n + 4 \)[/tex] : [tex]\( [5, 6, 7, 8, 9] \)[/tex]
2. [tex]\( a_n = 2n - 1 \)[/tex] : [tex]\( [1, 3, 5, 7, 9] \)[/tex]
3. [tex]\( a_n = 12 - 3n \)[/tex] : [tex]\( [9, 6, 3, 0, -3] \)[/tex]
4. [tex]\( a_n = 3^n \)[/tex] : [tex]\( [3, 9, 27, 81, 243] \)[/tex]
5. [tex]\( a_n = (-2)^n \)[/tex] : [tex]\( [-2, 4, -8, 16, -32] \)[/tex]
1. For the sequence [tex]\( a_n = n + 4 \)[/tex]:
- For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = 1 + 4 = 5 \][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = 2 + 4 = 6 \][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = 3 + 4 = 7 \][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = 4 + 4 = 8 \][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = 5 + 4 = 9 \][/tex]
So, the first 5 terms are: [tex]\( [5, 6, 7, 8, 9] \)[/tex].
2. For the sequence [tex]\( a_n = 2n - 1 \)[/tex]:
- For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = 2 \cdot 1 - 1 = 1 \][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = 2 \cdot 2 - 1 = 3 \][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = 2 \cdot 3 - 1 = 5 \][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = 2 \cdot 4 - 1 = 7 \][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = 2 \cdot 5 - 1 = 9 \][/tex]
So, the first 5 terms are: [tex]\( [1, 3, 5, 7, 9] \)[/tex].
3. For the sequence [tex]\( a_n = 12 - 3n \)[/tex]:
- For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = 12 - 3 \cdot 1 = 9 \][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = 12 - 3 \cdot 2 = 6 \][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = 12 - 3 \cdot 3 = 3 \][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = 12 - 3 \cdot 4 = 0 \][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = 12 - 3 \cdot 5 = -3 \][/tex]
So, the first 5 terms are: [tex]\( [9, 6, 3, 0, -3] \)[/tex].
4. For the sequence [tex]\( a_n = 3^n \)[/tex]:
- For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = 3^1 = 3 \][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = 3^2 = 9 \][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = 3^3 = 27 \][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = 3^4 = 81 \][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = 3^5 = 243 \][/tex]
So, the first 5 terms are: [tex]\( [3, 9, 27, 81, 243] \)[/tex].
5. For the sequence [tex]\( a_n = (-2)^n \)[/tex]:
- For [tex]\( n = 1 \)[/tex]:
[tex]\[ a_1 = (-2)^1 = -2 \][/tex]
- For [tex]\( n = 2 \)[/tex]:
[tex]\[ a_2 = (-2)^2 = 4 \][/tex]
- For [tex]\( n = 3 \)[/tex]:
[tex]\[ a_3 = (-2)^3 = -8 \][/tex]
- For [tex]\( n = 4 \)[/tex]:
[tex]\[ a_4 = (-2)^4 = 16 \][/tex]
- For [tex]\( n = 5 \)[/tex]:
[tex]\[ a_5 = (-2)^5 = -32 \][/tex]
So, the first 5 terms are: [tex]\( [-2, 4, -8, 16, -32] \)[/tex].
In summary, the first 5 terms for each sequence are:
1. [tex]\( a_n = n + 4 \)[/tex] : [tex]\( [5, 6, 7, 8, 9] \)[/tex]
2. [tex]\( a_n = 2n - 1 \)[/tex] : [tex]\( [1, 3, 5, 7, 9] \)[/tex]
3. [tex]\( a_n = 12 - 3n \)[/tex] : [tex]\( [9, 6, 3, 0, -3] \)[/tex]
4. [tex]\( a_n = 3^n \)[/tex] : [tex]\( [3, 9, 27, 81, 243] \)[/tex]
5. [tex]\( a_n = (-2)^n \)[/tex] : [tex]\( [-2, 4, -8, 16, -32] \)[/tex]
