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Sagot :
To determine which summation formula accurately represents the series [tex]\(13, 9, 5, 1\)[/tex], let's analyze each potential formula step by step.
### Option 1: [tex]\(\sum_{n=1}^4 (-4n + 17)\)[/tex]
Let's calculate the terms for [tex]\(n = 1, 2, 3, 4\)[/tex]:
- For [tex]\(n = 1\)[/tex]: [tex]\(-4(1) + 17 = -4 + 17 = 13\)[/tex]
- For [tex]\(n = 2\)[/tex]: [tex]\(-4(2) + 17 = -8 + 17 = 9\)[/tex]
- For [tex]\(n = 3\)[/tex]: [tex]\(-4(3) + 17 = -12 + 17 = 5\)[/tex]
- For [tex]\(n = 4\)[/tex]: [tex]\(-4(4) + 17 = -16 + 17 = 1\)[/tex]
The terms calculated are [tex]\(13, 9, 5, 1\)[/tex], which exactly match the given series.
### Option 2: [tex]\(\sum_{n=1}^4 (-4n - 15)\)[/tex]
Let's calculate the terms for [tex]\(n = 1, 2, 3, 4\)[/tex]:
- For [tex]\(n = 1\)[/tex]: [tex]\(-4(1) - 15 = -4 - 15 = -19\)[/tex]
- For [tex]\(n = 2\)[/tex]: [tex]\(-4(2) - 15 = -8 - 15 = -23\)[/tex]
- For [tex]\(n = 3\)[/tex]: [tex]\(-4(3) - 15 = -12 - 15 = -27\)[/tex]
- For [tex]\(n = 4\)[/tex]: [tex]\(-4(4) - 15 = -16 - 15 = -31\)[/tex]
The terms calculated are [tex]\(-19, -23, -27, -31\)[/tex], which do not match the given series.
### Option 3: [tex]\(\sum_{n=13}^{16} (n - 4)\)[/tex]
Let's calculate the terms for [tex]\(n = 13, 14, 15, 16\)[/tex]:
- For [tex]\(n = 13\)[/tex]: [tex]\(13 - 4 = 9\)[/tex]
- For [tex]\(n = 14\)[/tex]: [tex]\(14 - 4 = 10\)[/tex]
- For [tex]\(n = 15\)[/tex]: [tex]\(15 - 4 = 11\)[/tex]
- For [tex]\(n = 16\)[/tex]: [tex]\(16 - 4 = 12\)[/tex]
The terms calculated are [tex]\(9, 10, 11, 12\)[/tex], which do not match the given series.
### Option 4: [tex]\(\sum_{n=17}^{20} (n - 4)\)[/tex]
Let's calculate the terms for [tex]\(n = 17, 18, 19, 20\)[/tex]:
- For [tex]\(n = 17\)[/tex]: [tex]\(17 - 4 = 13\)[/tex]
- For [tex]\(n = 18\)[/tex]: [tex]\(18 - 4 = 14\)[/tex]
- For [tex]\(n = 19\)[/tex]: [tex]\(19 - 4 = 15\)[/tex]
- For [tex]\(n = 20\)[/tex]: [tex]\(20 - 4 = 16\)[/tex]
The terms calculated are [tex]\(13, 14, 15, 16\)[/tex], which do not match the given series.
### Conclusion
After evaluating each option, the correct summation formula that represents the series [tex]\(13, 9, 5, 1\)[/tex] is:
[tex]\[ \sum_{n=1}^4 (-4n + 17) \][/tex]
This matches the given series exactly.
### Option 1: [tex]\(\sum_{n=1}^4 (-4n + 17)\)[/tex]
Let's calculate the terms for [tex]\(n = 1, 2, 3, 4\)[/tex]:
- For [tex]\(n = 1\)[/tex]: [tex]\(-4(1) + 17 = -4 + 17 = 13\)[/tex]
- For [tex]\(n = 2\)[/tex]: [tex]\(-4(2) + 17 = -8 + 17 = 9\)[/tex]
- For [tex]\(n = 3\)[/tex]: [tex]\(-4(3) + 17 = -12 + 17 = 5\)[/tex]
- For [tex]\(n = 4\)[/tex]: [tex]\(-4(4) + 17 = -16 + 17 = 1\)[/tex]
The terms calculated are [tex]\(13, 9, 5, 1\)[/tex], which exactly match the given series.
### Option 2: [tex]\(\sum_{n=1}^4 (-4n - 15)\)[/tex]
Let's calculate the terms for [tex]\(n = 1, 2, 3, 4\)[/tex]:
- For [tex]\(n = 1\)[/tex]: [tex]\(-4(1) - 15 = -4 - 15 = -19\)[/tex]
- For [tex]\(n = 2\)[/tex]: [tex]\(-4(2) - 15 = -8 - 15 = -23\)[/tex]
- For [tex]\(n = 3\)[/tex]: [tex]\(-4(3) - 15 = -12 - 15 = -27\)[/tex]
- For [tex]\(n = 4\)[/tex]: [tex]\(-4(4) - 15 = -16 - 15 = -31\)[/tex]
The terms calculated are [tex]\(-19, -23, -27, -31\)[/tex], which do not match the given series.
### Option 3: [tex]\(\sum_{n=13}^{16} (n - 4)\)[/tex]
Let's calculate the terms for [tex]\(n = 13, 14, 15, 16\)[/tex]:
- For [tex]\(n = 13\)[/tex]: [tex]\(13 - 4 = 9\)[/tex]
- For [tex]\(n = 14\)[/tex]: [tex]\(14 - 4 = 10\)[/tex]
- For [tex]\(n = 15\)[/tex]: [tex]\(15 - 4 = 11\)[/tex]
- For [tex]\(n = 16\)[/tex]: [tex]\(16 - 4 = 12\)[/tex]
The terms calculated are [tex]\(9, 10, 11, 12\)[/tex], which do not match the given series.
### Option 4: [tex]\(\sum_{n=17}^{20} (n - 4)\)[/tex]
Let's calculate the terms for [tex]\(n = 17, 18, 19, 20\)[/tex]:
- For [tex]\(n = 17\)[/tex]: [tex]\(17 - 4 = 13\)[/tex]
- For [tex]\(n = 18\)[/tex]: [tex]\(18 - 4 = 14\)[/tex]
- For [tex]\(n = 19\)[/tex]: [tex]\(19 - 4 = 15\)[/tex]
- For [tex]\(n = 20\)[/tex]: [tex]\(20 - 4 = 16\)[/tex]
The terms calculated are [tex]\(13, 14, 15, 16\)[/tex], which do not match the given series.
### Conclusion
After evaluating each option, the correct summation formula that represents the series [tex]\(13, 9, 5, 1\)[/tex] is:
[tex]\[ \sum_{n=1}^4 (-4n + 17) \][/tex]
This matches the given series exactly.
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