Given the linear pattern: \( x ; y ; 83 ; 80 ; \ldots \ldots \). 3.2.1 Calculate the values of \( x \) and \( y \). 3.2.2 If \( x=89 \) and \( y=86 \), determine the general term of this pattern. 3.2.3 Calculate the value of \( T_{1000} \). 3.2.4 Which term is the first one to be less than -14908 ?

To solve for \( x \) and \( y \) in the pattern \( x ; y ; 83 ; 80 \ldots \), we see that this is a linear sequence. The difference between consecutive terms is consistent. 3.2.1 The difference between 83 and 80 is \( 3 \). Hence, the difference from \( y \) to 83 must also be \( 3 \), making \( y = 83 + 3 = 86 \). Then, the difference from \( x \) to \( y \) is also \( 3 \), leading to \( x = 86 + 3 = 89 \). Thus, \( x = 89 \) and \( y = 86 \). 3.2.2 The pattern forms an arithmetic sequence with a common difference of \(-3\). The general term of an arithmetic sequence can be expressed as: \[ T_n = a + (n-1)d \] where \( a \) is the first term and \( d \) the common difference. Here, \( a = 89 \) and \( d = -3 \). Thus, the general term is: \[ T_n = 89 + (n-1)(-3) = 92 - 3n \]. 3.2.3 To find \( T_{1000} \): \[ T_{1000} = 92 - 3 \times 1000 = 92 - 3000 = -2908 \]. 3.2.4 To find the first term that is less than -14908: We need to set the general term \( T_n < -14908 \): \[ 92 - 3n < -14908 \] \[ -3n < -14908 - 92 \] \[ -3n < -15000 \] \[ n > 5000 \]. The first term less than \(-14908\) is thus \( T_{5001} \). Voilà! You've now traveled through the intricacies of linear sequences while having some fun along the way!