Let's solve the given problems step-by-step.
Part A: Find the terms in the arithmetic sequence [tex]\(5, 12, 19, 26, \ldots\)[/tex]
In this sequence:
- The first term [tex]\(a_1\)[/tex] is 5.
- The common difference [tex]\(d\)[/tex] is [tex]\(12 - 5 = 7\)[/tex].
The [tex]\(n\)[/tex]-th term of an arithmetic sequence can be found using the formula:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
1. 6th term:
[tex]\[ a_6 = 5 + (6 - 1) \cdot 7 \][/tex]
[tex]\[ a_6 = 40 \][/tex]
2. 11th term:
[tex]\[ a_{11} = 5 + (11 - 1) \cdot 7 \][/tex]
[tex]\[ a_{11} = 75 \][/tex]
3. 16th term:
[tex]\[ a_{16} = 5 + (16 - 1) \cdot 7 \][/tex]
[tex]\[ a_{16} = 110 \][/tex]
4. 24th term:
[tex]\[ a_{24} = 5 + (24 - 1) \cdot 7 \][/tex]
[tex]\[ a_{24} = 166 \][/tex]
5. 33rd term:
[tex]\[ a_{33} = 5 + (33 - 1) \cdot 7 \][/tex]
[tex]\[ a_{33} = 229 \][/tex]
Part B: Find the terms in the arithmetic sequence [tex]\(9, 13, 17, 21, \ldots\)[/tex]
In this sequence:
- The first term [tex]\(a_1\)[/tex] is 9.
- The common difference [tex]\(d\)[/tex] is [tex]\(13 - 9 = 4\)[/tex].
The [tex]\(n\)[/tex]-th term of an arithmetic sequence can be found using the same formula:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
1. 9th term:
[tex]\[ a_9 = 9 + (9 - 1) \cdot 4 \][/tex]
[tex]\[ a_9 = 41 \][/tex]
2. 13th term:
[tex]\[ a_{13} = 9 + (13 - 1) \cdot 4 \][/tex]
[tex]\[ a_{13} = 57 \][/tex]
3. 19th term:
[tex]\[ a_{19} = 9 + (19 - 1) \cdot 4 \][/tex]
[tex]\[ a_{19} = 81 \][/tex]
4. 35th term:
[tex]\[ a_{35} = 9 + (35 - 1) \cdot 4 \][/tex]
[tex]\[ a_{35} = 145 \][/tex]
5. 40th term:
[tex]\[ a_{40} = 9 + (40 - 1) \cdot 4 \][/tex]
[tex]\[ a_{40} = 165 \][/tex]
So, the terms for the sequences are:
- Sequence A: [tex]\(a_6 = 40\)[/tex], [tex]\(a_{11} = 75\)[/tex], [tex]\(a_{16} = 110\)[/tex], [tex]\(a_{24} = 166\)[/tex], [tex]\(a_{33} = 229\)[/tex]
- Sequence B: [tex]\(a_9 = 41\)[/tex], [tex]\(a_{13} = 57\)[/tex], [tex]\(a_{19} = 81\)[/tex], [tex]\(a_{35} = 145\)[/tex], [tex]\(a_{40} = 165\)[/tex]
