Sure, let's solve each sequence step-by-step:
### 1. Sequence: [tex]\(28, 24, 20, 16, 12, 8\)[/tex]
We need to find the 16th term of this sequence.
Step-by-step:
1. Identify that this is an arithmetic sequence where each term decreases by a common difference of -4.
2. The first term ([tex]\(a_1\)[/tex]) is 28.
3. The common difference ([tex]\(d\)[/tex]) is -4.
4. We need to find the 16th term ([tex]\(T_{16}\)[/tex]).
The formula for the [tex]\(n\)[/tex]th term [tex]\(T_n\)[/tex] of an arithmetic sequence is:
[tex]\[ T_n = a_1 + (n - 1) \cdot d \][/tex]
Plugging in the values, we get:
[tex]\[ T_{16} = 28 + (16 - 1) \cdot (-4) \][/tex]
[tex]\[ T_{16} = 28 + 15 \cdot (-4) \][/tex]
[tex]\[ T_{16} = 28 - 60 \][/tex]
[tex]\[ T_{16} = -32 \][/tex]
Thus, the 16th term of the sequence [tex]\(28, 24, 20, 16, 12, 8\)[/tex] is [tex]\(-32\)[/tex].
### 2. Sequence: [tex]\(-3, -6, -9, -12, -15\)[/tex]
We need to find the 10th term of this sequence.
Step-by-step:
1. Recognize this as an arithmetic sequence with a common difference of -3.
2. The first term ([tex]\(a_1\)[/tex]) is -3.
3. The common difference ([tex]\(d\)[/tex]) is -3.
4. We need to find the 10th term ([tex]\(T_{10}\)[/tex]).
Using the same [tex]\(n\)[/tex]th term formula:
[tex]\[ T_{10} = -3 + (10 - 1) \cdot (-3) \][/tex]
[tex]\[ T_{10} = -3 + 9 \cdot (-3) \][/tex]
[tex]\[ T_{10} = -3 - 27 \][/tex]
[tex]\[ T_{10} = -30 \][/tex]
Thus, the 10th term of the sequence [tex]\(-3, -6, -9, -12, -15\)[/tex] is [tex]\(-30\)[/tex].
### 3. Sequence: [tex]\(14, x+28, 35, 43\)[/tex]
We need to find the 25th term of this sequence.
Step-by-step:
1. Identify that this is an arithmetic sequence. We need to determine the common difference [tex]\(d\)[/tex] and the unknown [tex]\(x\)[/tex].
2. The first term ([tex]\(a_1\)[/tex]) is 14.
3. The third term ([tex]\(T_3\)[/tex]) is 35. Since the sequence is arithmetic, we have:
[tex]\[35 - 14 = 21\][/tex]
[tex]\[\text{This 21 is a result of 2 common differences } 2d\][/tex]
[tex]\[ d = \frac{21}{2} = 10.5\][/tex]
Using this common difference, we can check the second and fourth terms:
[tex]\[ \text{Second term} = 14 + 10.5 = 24.5 \neq x + 28 \][/tex]
Thus, [tex]\(24.5 = x + 28 \implies x = -3.5\)[/tex].
4. Finally, to find the 25th term ([tex]\(T_{25}\)[/tex]):
[tex]\[ T_{25} = 14 + (25 - 1) \cdot 10.5\][/tex]
[tex]\[ T_{25} = 14 + 24 \cdot 10.5 \][/tex]
[tex]\[ T_{25} = 14 + 252 \][/tex]
[tex]\[ T_{25} = 266 \][/tex]
Thus, the 25th term of the sequence [tex]\(14, x+28, 35, 43\)[/tex] where [tex]\(x = -3.5\)[/tex] is [tex]\(266\)[/tex].
Based on the steps followed, the final results are:
1. 16th term of the first sequence is [tex]\(-32\)[/tex].
2. 10th term of the second sequence is [tex]\(-30\)[/tex].
3. 25th term of the third sequence is [tex]\(267\)[/tex].
