\begin{tabular}{l} culate the sum of each of the following arithmetic series: \\ \( 7+9+11+\ldots+105 \) \\ \( 2-3-8-\ldots-368 \) \\ (2) \( 50+43+36+\ldots-643 \) \\ \( 2+\frac{13}{4}+\frac{9}{2}+\ldots+82 \) \\ \( -2 p+2 p+6 p+\ldots+102 p \) \\ (4) \( -15-12-9-\ldots+432 \) \\ \\ \hline\end{tabular}

To sum each of the given arithmetic series, we will apply the formula for the sum of an arithmetic series: \[ S_n = \frac{n}{2} (a + l) \] where \( n \) is the number of terms, \( a \) is the first term, and \( l \) is the last term. 1. For \( 7 + 9 + 11 + \ldots + 105 \): - First term \( a = 7 \), last term \( l = 105 \), and the common difference \( d = 2 \). - Number of terms \( n = \frac{(l - a)}{d} + 1 = \frac{(105 - 7)}{2} + 1 = 50 \). - Sum \( S_{50} = \frac{50}{2} (7 + 105) = 25 \times 112 = 2800 \). 2. For \( 2 - 3 - 8 - \ldots - 368 \): - First term \( a = 2 \), last term \( l = -368 \), and common difference \( d = -5 \). - \( n = \frac{(l - a)}{d} + 1 = \frac{(-368 - 2)}{-5} + 1 = 74 \). - Sum \( S_{74} = \frac{74}{2} (2 - 368) = 37 \times (-366) = -13642 \). 3. For \( 50 + 43 + 36 + \ldots - 643 \): - First term \( a = 50 \), last term \( l = -643 \), common difference \( d = -7 \). - \( n = \frac{(l - a)}{d} + 1 = \frac{(-643 - 50)}{-7} + 1 = 100 \). - Sum \( S_{100} = \frac{100}{2} (50 - 643) = 50 \times (-593) = -29650 \). 4. For \( 2 + \frac{13}{4} + \frac{9}{2} + \ldots + 82 \): - First term \( a = 2\), last term \( l = 82 \), converting \( \frac{9}{2} = 4.5 \) to decimal gives a common difference \( d = \frac{29}{4} \). - \( n = \frac{(l - a)}{d} + 1 = \frac{(82 - 2)}{\frac{29}{4}} + 1 = 14 \). - Sum \( S_{14} = \frac{14}{2} (2 + 82) = 7 \times 84 = 588 \). 5. For \( -2p + 2p + 6p + \ldots + 102p \): - First term \( a = -2p \), last term \( l = 102p \), and common difference \( d = 4p \). - \( n = \frac{(l - a)}{d} + 1 = \frac{(102p + 2p)}{4p} + 1 = 26 \). - Sum \( S_{26} = \frac{26}{2} (-2 + 102)p = 13 \times 100p = 1300p \). 6. For \( -15 - 12 - 9 - \ldots + 432 \): - First term \( a = -15 \), last term \( l = 432 \), common difference \( d = 3 \). - \( n = \frac{(l - a)}{d} + 1 = \frac{(432 + 15)}{3} + 1 = 149 \). - Sum \( S_{149} = \frac{149}{2} (-15 + 432) = 74.5 \times 417 = 31046.5 \). Thus, the final sums of the series are: 1. 2800 2. -13642 3. -29650 4. 588 5. \( 1300p \) 6. 31046.5