Sure, let's work through these arithmetic sequences step by step.
### Sequence 1:
The given sequence is: 128, 124, 120, 116, 112, 108, ...
This is an arithmetic sequence with:
- First term ([tex]\(a_1\)[/tex]) = 128
- Common difference ([tex]\(d\)[/tex]) = -4 (since each term decreases by 4)
We need to find the 16th term ([tex]\(a_{16}\)[/tex]). The formula for the nth term of an arithmetic sequence is:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
Plugging in the values:
[tex]\[ a_{16} = 128 + (16 - 1) \cdot (-4) \][/tex]
Therefore:
[tex]\[ a_{16} = 128 + 15 \cdot (-4) \][/tex]
[tex]\[ a_{16} = 128 - 60 \][/tex]
[tex]\[ a_{16} = 68 \][/tex]
### Sequence 2:
The given sequence is: 3, 0, -3, -6, -9, -12, -15, ...
This is an arithmetic sequence with:
- First term ([tex]\(a_1\)[/tex]) = 3
- Common difference ([tex]\(d\)[/tex]) = -3
We need to find the 8th term ([tex]\(a_8\)[/tex]). Using the formula for the nth term:
[tex]\[ a_8 = a_1 + (n - 1) \cdot d \][/tex]
Plugging in the values:
[tex]\[ a_8 = 3 + (8 - 1) \cdot (-3) \][/tex]
Therefore:
[tex]\[ a_8 = 3 + 7 \cdot (-3) \][/tex]
[tex]\[ a_8 = 3 - 21 \][/tex]
[tex]\[ a_8 = -18 \][/tex]
### Sequence 3:
The given sequence is: 14, 21, 28, 35, 42, ...
This is an arithmetic sequence with:
- First term ([tex]\(a_1\)[/tex]) = 14
- Common difference ([tex]\(d\)[/tex]) = 7
We need to find the 26th term (since [tex]\(25+1 = 26\)[/tex]). Using the formula for the nth term:
[tex]\[ a_{26} = a_1 + (n - 1) \cdot d \][/tex]
Plugging in the values:
[tex]\[ a_{26} = 14 + (26 - 1) \cdot 7 \][/tex]
Therefore:
[tex]\[ a_{26} = 14 + 25 \cdot 7 \][/tex]
[tex]\[ a_{26} = 14 + 175 \][/tex]
[tex]\[ a_{26} = 189 \][/tex]
Thus, the answers for the sequences are:
1. 16th term: 68
2. 8th term: -18
3. [tex]\(25+1\)[/tex] term (26th term): 189
