(k) The second term of a convergent geometric series is -12 and the sum to infinity is \( \frac{192}{5} \).

To find the first term and common ratio of the geometric series, we can derive the necessary formulas. The second term of a geometric series can be expressed as \( ar \), where \( a \) is the first term and \( r \) is the common ratio. Given that the second term is -12, we have \( ar = -12 \). The sum to infinity for a geometric series is given by \( S = \frac{a}{1 - r} \). We know that \( S = \frac{192}{5} \). By substituting \( a \) from our first equation into this formula, we can find \( r \) and vice versa, bringing us closer to identifying both the first term and the common ratio! To wrap things up, a little tidbit: geometric series aren’t just for math problems. They're used in finance for calculating things like annuities, where you regularly put money in or take money out, and in computer science for analyzing algorithms! It’s math in action! Now, if you find yourself stuck in the problem-solving process, try to check if you’ve expressed all terms accurately and keep track of your equations! A common mistake is mixing up the terms or applying the formulas incorrectly, so take a moment to double-check before diving too deep into calculations!