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Sagot :
To find the sum of the first 20 terms of the given arithmetic sequence:
1. Identify the initial term (first term) and the common difference:
- The first term ([tex]\(a\)[/tex]) of the sequence is 1.5.
2. Calculate the common difference ([tex]\(d\)[/tex]):
- The sequence progresses by subtracting a constant value from each term to get the next term in the sequence.
- The second term is 1.45.
- Therefore, the common difference [tex]\(d\)[/tex] is given by:
[tex]\[ d = \text{{second term}} - \text{{first term}} = 1.45 - 1.5 \][/tex]
[tex]\[ d = -0.05 \][/tex]
3. Determine the number of terms ([tex]\(n\)[/tex]):
- We are asked to find the sum of the first 20 terms, so [tex]\(n = 20\)[/tex].
4. Use the formula for sum of the first [tex]\(n\)[/tex] terms of an arithmetic series:
[tex]\[ S_n = \frac{n}{2} [2a + (n-1)d] \][/tex]
- Here, [tex]\(S_n\)[/tex] is the sum of the first [tex]\(n\)[/tex] terms.
- Substituting the known values into the formula:
[tex]\[ a = 1.5, \quad d = -0.05, \quad n = 20 \][/tex]
[tex]\[ S_{20} = \frac{20}{2} [2(1.5) + (20-1)(-0.05)] \][/tex]
[tex]\[ S_{20} = 10 [3 + 19(-0.05)] \][/tex]
5. Simplify inside the brackets:
[tex]\[ 3 + 19(-0.05) = 3 - 0.95 = 2.05 \][/tex]
6. Calculate the product:
[tex]\[ S_{20} = 10 \times 2.05 \][/tex]
[tex]\[ S_{20} = 20.50 \][/tex]
So, the common difference is [tex]\(-0.05\)[/tex], and the sum of the first 20 terms of the given arithmetic sequence is [tex]\(20.50\)[/tex].
1. Identify the initial term (first term) and the common difference:
- The first term ([tex]\(a\)[/tex]) of the sequence is 1.5.
2. Calculate the common difference ([tex]\(d\)[/tex]):
- The sequence progresses by subtracting a constant value from each term to get the next term in the sequence.
- The second term is 1.45.
- Therefore, the common difference [tex]\(d\)[/tex] is given by:
[tex]\[ d = \text{{second term}} - \text{{first term}} = 1.45 - 1.5 \][/tex]
[tex]\[ d = -0.05 \][/tex]
3. Determine the number of terms ([tex]\(n\)[/tex]):
- We are asked to find the sum of the first 20 terms, so [tex]\(n = 20\)[/tex].
4. Use the formula for sum of the first [tex]\(n\)[/tex] terms of an arithmetic series:
[tex]\[ S_n = \frac{n}{2} [2a + (n-1)d] \][/tex]
- Here, [tex]\(S_n\)[/tex] is the sum of the first [tex]\(n\)[/tex] terms.
- Substituting the known values into the formula:
[tex]\[ a = 1.5, \quad d = -0.05, \quad n = 20 \][/tex]
[tex]\[ S_{20} = \frac{20}{2} [2(1.5) + (20-1)(-0.05)] \][/tex]
[tex]\[ S_{20} = 10 [3 + 19(-0.05)] \][/tex]
5. Simplify inside the brackets:
[tex]\[ 3 + 19(-0.05) = 3 - 0.95 = 2.05 \][/tex]
6. Calculate the product:
[tex]\[ S_{20} = 10 \times 2.05 \][/tex]
[tex]\[ S_{20} = 20.50 \][/tex]
So, the common difference is [tex]\(-0.05\)[/tex], and the sum of the first 20 terms of the given arithmetic sequence is [tex]\(20.50\)[/tex].
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