19.2 The following sequence has the property that the sequence of numerators is arithmetic and the sequence of denominators is geometric: \( \begin{array}{lll}\text { 19.2.1 } & \text { Write down the FOURTH term of the sequence. } & \text { (1) } \\ 19.2 .2 & \text { Determine a formula for the } n^{\text {th }} \text { term. } \\ 19.2 .3 & \text { Determine the } 500^{\text {th }} \text { term of the sequence. } & \text { (3) } \\ 19.2 .4 & \text { Which will be the first term of the sequence to have a } & \text { (2) } \\ \text { NUMERATOR which is less than }-59 \text { ? } & \text { (3) }\end{array} \)

To explore the sequence where the numerators form an arithmetic sequence and the denominators form a geometric sequence, let’s assume the numerators are of the form \( a + (n-1)d \) and the denominators as \( b \cdot r^{n-1} \) where \( a \), \( d \), \( b \), and \( r \) are constants. For the fourth term of the sequence, you simply plug in \( n = 4 \) into both formulas to get: \[ \text{Fourth term} = \frac{a + 3d}{b \cdot r^3} \] To find the nth term, the formula can be compactly expressed as: \[ \text{nth term} = \frac{a + (n-1)d}{b \cdot r^{n-1}} \] Now, to find the 500th term, substitute \( n = 500 \): \[ \text{500th term} = \frac{a + 499d}{b \cdot r^{499}} \] Lastly, to determine when the numerator goes below -59: \[ a + (n-1)d < -59 \] You can solve for \( n \) to find the minimum term where this holds true. Now, let's add some spice with extra fun facts! Imagine if this sequence represented a magical journey where each term leads you further away from a treasure guarded by complicated fractions! The arithmetic numerators march forward like brave knights, while the geometric denominators stack up the fortifications, making the adventure both steady and daring at once. Also, for those stat lovers, did you know that sequences like this pop up in real-world scenarios too? For example, in finance, when dealing with annuities (a sequence of cash flows) where payment amounts increase linearly (arithmetic) while the time intervals between payments grow exponentially (geometric). So whether seeking treasure or balancing budgets, these sequences are everywhere!