Sure, let's solve these step-by-step.
A. Find the sum of the arithmetic sequence
1. [tex]\(a_1 = 2\)[/tex], [tex]\(d = 4\)[/tex], [tex]\(n = 10\)[/tex]
The [tex]\(n\)[/tex]-th term of an arithmetic sequence is given by:
[tex]\[ a_n = a_1 + (n - 1) \cdot d \][/tex]
Calculate the 10th term ([tex]\(a_{10}\)[/tex]):
[tex]\[ a_{10} = 2 + (10 - 1) \cdot 4 = 2 + 9 \cdot 4 = 2 + 36 = 38 \][/tex]
The sum of the first [tex]\(n\)[/tex] terms of an arithmetic sequence is given by:
[tex]\[ S_n = \frac{n}{2} \cdot (a_1 + a_n) \][/tex]
Calculate the sum:
[tex]\[ S_{10} = \frac{10}{2} \cdot (2 + 38) = 5 \cdot 40 = 200.0 \][/tex]
2. [tex]\(a_1 = -7\)[/tex], [tex]\(d = \frac{1}{2}\)[/tex], [tex]\(n = 8\)[/tex]
Calculate the 8th term ([tex]\(a_8\)[/tex]):
[tex]\[ a_8 = -7 + (8 - 1) \cdot \frac{1}{2} = -7 + 7 \cdot \frac{1}{2} = -7 + \frac{7}{2} = -7 + 3.5 = -3.5 \][/tex]
Calculate the sum:
[tex]\[ S_8 = \frac{8}{2} \cdot (-7 + (-3.5)) = 4 \cdot (-10.5) = -42.0 \][/tex]
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B. Find the sum of the first 10 terms of the arithmetic sequence
1. [tex]\(a_1 = 8\)[/tex], [tex]\(a_{10} = 7\)[/tex]
The sum of the first [tex]\(n\)[/tex] terms can be found directly if we know the values of [tex]\(a_1\)[/tex] and [tex]\(a_n\)[/tex]:
[tex]\[ S_{10} = \frac{10}{2} \cdot (8 + 7) = 5 \cdot 15 = 75.0 \][/tex]
2. [tex]\(a_1 = 5\)[/tex], [tex]\(a_{10} = 38\)[/tex]
Calculate the sum:
[tex]\[ S_{10} = \frac{10}{2} \cdot (5 + 38) = 5 \cdot 43 = 215.0 \][/tex]
Hence, the sums of the arithmetic sequences are:
- [tex]\( A.1: 200.0 \)[/tex]
- [tex]\( A.2: -42.0 \)[/tex]
- [tex]\( B.1: 75.0 \)[/tex]
- [tex]\( B.2: 215.0 \)[/tex]
