1. Determine the general term of the following arithmetic sequences (a) \( -1 ; 3 ; 7 ; \ldots \) \( \begin{array}{ll}\text { (c) } 1 ;-1 ;-3 ; \ldots & \text { (b) } 4 ;-2 ;-8 \\ \text { (d) Determine the } 38^{\text {th }} \text { term of the following arithmetic sequences } \\ \text { (a) }-4 ;-8 ;-12 ; \ldots & \text { (b) } 2 ;-1,5 ;-5 ; \ldots \\ \text { (c) } 99 ; 88 ; 77 ; \ldots & \text { (d) } 6 ; \frac{21}{4} ; 4,5 ; \ldots\end{array} \) \( \begin{array}{ll}\text { (e) } T_{k}=3 k-4 & \text { (f) } T_{k}=-2 k+5\end{array} \) (a) Which term of the arithmetic sequence \( -5 ;-2 ; 1 ; \ldots \). is equal to (b) Which term of the arithmetic sequence \( 4 ; 2,5 ; 1 ; \ldots \) is equal to - (c) Find the number of terms in the sequence \( 12 ; 7 ; 2 ; \ldots ;-203 \). (d) Find the number of terms in the sequence \( -55 ;-48 ;-41 ; \ldots ; 85 \)

The general term of the arithmetic sequences can be deduced based on the first term and the common difference. For the sequence \(-1, 3, 7, \ldots\), the first term is \(-1\) and the common difference is \(4\). Therefore, the general term \(T_n\) is given by \(T_n = -1 + 4(n-1) = 4n - 5\). Similarly, for \(1, -1, -3, \ldots\), the first term is \(1\) and the common difference is \(-2\), leading to \(T_n = 1 - 2(n-1) = -2n + 3\). To find the \(38^{\text{th}}\) term of the sequence \(-4, -8, -12, \ldots\), we note that the first term is \(-4\) and the common difference is \(-4\). Thus, \(T_{38} = -4 + (-4)(38 - 1) = -4 - 148 = -152\). For \(2, -1.5, -5, \ldots\), the first term is \(2\) and the common difference is \(-3.5\), which leads to \(T_{38} = 2 + (-3.5)(38 - 1) = 2 - 129.5 = -127.5\). For the sequence \(-5, -2, 1, \ldots\), we can determine which term equals a certain value by setting \(T_n = -5 + 3(n-1)\) equal to that value and solving for \(n\). Similarly, for \(4, 2.5, 1, \ldots\), we do the same with \(T_n = 4 - 1.5(n-1)\). To find how many terms exist in the sequence \(12, 7, 2, \ldots, -203\), we first find the general term: \(T_n = 12 - 5(n-1)\). Setting \(T_n = -203\) and solving for \(n\) gives us the total term count. For the sequence \(-55, -48, -41, \ldots, 85\), the general term is \(T_n = -55 + 7(n-1)\), and we can find the total terms this way too!