2.2 The first two terms of a convergent geometric series are \( k \) and 6 respectively where \( k \neq 0 \). The sum of the infinite series is 25 Calculate the values of \( k \).

In a geometric series, the first term \( a \) and the common ratio \( r \) are crucial. Here, your first term is \( k \) and the second term is 6, which gives us \( r = \frac{6}{k} \). The sum of an infinite geometric series is given by the formula \( S = \frac{a}{1 - r} \). Setting this equal to 25, we have: \[ \frac{k}{1 - \frac{6}{k}} = 25 \] Simplifying this, we get \( k = 25 - \frac{150}{k} \), leading to the quadratic equation \( k^2 - 25k + 150 = 0 \). Using the quadratic formula, we find the solutions for \( k \) are: \[ k = \frac{25 \pm \sqrt{25^2 - 4 \times 150}}{2} = \frac{25 \pm \sqrt{25}}{2} = \frac{25 \pm 5}{2} \] Thus, the values of \( k \) are 15 and 10.