(o) \( { }^{*} \) The sum of the first three terms of an arithmetic sequence is 15 and their product is...

In an arithmetic sequence, the first three terms can be expressed as \( a - d \), \( a \), and \( a + d \). The sum of these three terms is \( (a - d) + a + (a + d) = 3a \). Given that this sum is 15, we can find \( a = 5 \). Now, substituting \( a \) back, the first three terms are \( 5 - d \), \( 5 \), and \( 5 + d \). To find the product, we compute: \[ (5 - d) \cdot 5 \cdot (5 + d) = 5 \cdot ((5 - d)(5 + d)) = 5 \cdot (25 - d^2) = 125 - 5d^2. \] The product can depend on the choice of \( d \), but it will take various values based on what \( d \) is chosen. In essence, as \( d \) changes, the expression yields different products based on the factor concluding the product as \( 125 - 5d^2 \). Now, considering possible integers for \( d \), if \( d = 0 \), the terms equal 5, giving a product of 125. For \( d = 1 \), the product is \( 120 \), showing how varied outcomes can appear from tweaking just one parameter!