16. In an arithmetic sequence the 17 th term is 9 times the first term and t 9 th term is 6 less than 3 times the 3 rd term. Find the first 3 terms the sequence.

In an arithmetic sequence, the \( n \)-th term can be expressed as \( a_n = a + (n-1)d \), where \( a \) is the first term and \( d \) is the common difference. Given that the 17th term is 9 times the first term, we can write the equation: \( a + 16d = 9a \), or simplified, \( 16d = 8a \) (which implies \( d = \frac{1}{2}a \)). For the 9th term, which is said to be 6 less than 3 times the 3rd term, we set \( a + 8d = 3(a + 2d) - 6 \). Substituting \( d = \frac{1}{2}a \) into this equation provides a straightforward way to solve for \( a \), and ultimately, the first three terms. Upon solving, we find that \( a = 6 \) and \( d = 3 \). This results in the first three terms of the sequence being 6, 9, and 12! Isn’t it interesting how just a few equations can unravel an entire sequence?