Question: Which summation formula represents the series? 13 + 9 + 5 + 1 \[ \sum_{n=13}^{16} (n - 4) \]…

Which summation formula represents the series?

13 + 9 + 5 + 1

\[ \sum_{n=13}^{16} (n - 4) \]

\[ \sum_{n=17}^{20} (n - 4) \]

\[ \sum_{n=1}^{4} (-4n - 15) \]

\[ \sum_{n=1}^{4} (-4n + 17) \]

Solution

To determine which summation formula represents the series \(13 + 9 + 5 + 1\), let’s analyze each option. First, compute the series: \[ 13 + 9 + 5 + 1 = 28 \] Now, analyze each summation formula: 1. \(\sum_{n=13}^{16} (n-4)\) Calculate the terms: - For \(n=13\): \(13 - 4 = 9\) - For \(n=14\): \(14 - 4 = 10\) - For \(n=15\): \(15 - 4 = 11\) - For \(n=16\): \(16 - 4 = 12\) Sum: \(9 + 10 + 11 + 12 = 42\) 2. \(\sum_{n=17}^{20} (n-4)\) Calculate the terms: - For \(n=17\): \(17 - 4 = 13\) - For \(n=18\): \(18 - 4 = 14\) - For \(n=19\): \(19 - 4 = 15\) - For \(n=20\): \(20 - 4 = 16\) Sum: \(13 + 14 + 15 + 16 = 58\) 3. \(\sum_{n=1}^{4} (-4n + 17)\) Calculate the terms: - For \(n=1\): \(-4(1) + 17 = 13\) - For \(n=2\): \(-4(2) + 17 = 9\) - For \(n=3\): \(-4(3) + 17 = 5\) - For \(n=4\): \(-4(4) + 17 = 1\) Sum: \(13 + 9 + 5 + 1 = 28\) 4. \(\sum_{n=1}^{4} (-4n - 15)\) (This option seems unlikely given the trend) It appears that \(\sum_{n=1}^{4} (-4n + 17)\) matches the series \(13 + 9 + 5 + 1\).