3 Mathematics/control test 1 SBA task 2 Ehlanzeni QUESTION 1 1.1 Consider the following quadratic sequence: \( -2 ;-3 ;-6 ;-11 ; \ldots \) 1.1.1 The first differences also form a sequence. Write down the sequence. 1.1.2 Determine the general term of the first differences. 1.1.3 Calculate the difference between the \( 99^{\mathrm{th}} \) and the \( 100^{\mathrm{dh}} \) term of the quadratic sequence. 1.1.4 If the sum of the first differences is given by \( S_{k}=-k^{2} \). Calculate the fiftieth ( \( T_{50} \) ) of the quadratic sequence. \( 1.2 \quad 4-x ; 3 \) and \( \sqrt{2 x+3} \) are the first three terms of an arithmetic sequence. Determine the value of \( x \).
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The first differences of the quadratic sequence \(-2, -3, -6, -11, \ldots\) are obtained by subtracting consecutive terms. Therefore, the first differences are \(-1, -3, -5, \ldots\), indicating a linear change, which hints at the quadratic nature of the original sequence. To determine \( x \), we first note that in an arithmetic sequence, the difference between consecutive terms should remain constant. By setting \( (3 - (4 - x)) = (\sqrt{2x + 3} - 3) \) and solving for \( x \), we can ensure that our first three terms indeed have the equal spacing characteristic of arithmetic sequences.