QUESTION 2 2.1 The arithmetic sequence \( 4 ; 10 ; 16 ; \ldots \) is the sequence of the first difference of a quadratic sequence with a first term equal to 3 . Determine the \( 50^{\text {th }} \) term of the quadratic sequence. 2.2 Given the finite arithmetic sequence \( 5 ; 1 ;-3 ; \ldots-83 ;-87 \) 2.2.1 Calculate the number of terms in the sequence 2.2.2 Calculate the sum of all the negative numbers in the sequence.

To determine the \(50^{\text{th}}\) term of the quadratic sequence, we first find the differences in the arithmetic sequence \(4; 10; 16; \ldots\) which gives a consistent second difference of \(6\). Since the first term of the quadratic sequence is \(3\) with a second difference of \(6\), we use the quadratic formula \(a_n = a + (n-1)d + \frac{(n-1)(n-2)}{2} \cdot c\) where \(c\) is half the second difference. Plugging in the values, we find the \(50^{\text{th}}\) term is calculated as: \[ a_{50} = 3 + (50 - 1) \cdot 4 + \frac{(50-1)(50-2)}{2} \cdot 3 = 3 + 196 + 1176 = 1375. \] Turning to the arithmetic sequence \(5; 1; -3; \ldots; -83; -87\), we spot that it starts at \(5\) and decreases by \(4\) each time. To find the number of terms, we set the last term \( -87 = 5 + (n-1)(-4)\), solve for \(n\), yielding \(n = 23\). For the sum of all negative numbers in this sequence, the negative numbers start from \(-3\) to \(-87\). The sequence of negative terms is \(-3, -7, -11, \ldots, -87\) and is an arithmetic sequence with the first term of \(-3\) and \(d = -4\). To find the number of terms, we solve for \(n\) in \(-87 = -3 + (n-1)(-4)\), resulting in \(n = 22 - 1\) or \(n = 22\). The sum is calculated using the formula for the sum of an arithmetic series \(S_n = \frac{n}{2} (a + l)\), for \(a = -3\), \(l = -87\), and \(n = 22\), thus, \[ S = \frac{22}{2}\left(-3 - 87\right) = 11 \cdot (-90) = -990. \]