The following sequence forms a convergent geometric sequence: \[ 5 y ; y^{2} ; \frac{y^{3}}{5} ; \cdots \] a) Determine the possible values of \( y \). b) If \( y=2 \), Calculate \( \mathrm{S}_{\infty} \).

To determine the possible values of \( y \) for the sequence \( 5y, y^2, \frac{y^3}{5}, \cdots \) to be a convergent geometric sequence, we first need to find the common ratio \( r \). The common ratio can be found by dividing the second term by the first term and the third term by the second term: \[ r = \frac{y^2}{5y} = \frac{y}{5} \] \[ r = \frac{\frac{y^3}{5}}{y^2} = \frac{y}{5} \] Since both ratios are equal, we can deduce that the common ratio is consistent throughout the sequence. For the sequence to converge, the absolute value of the common ratio must be less than 1: \[ \left|\frac{y}{5}\right| < 1 \] This leads to two inequalities: \[ -1 < \frac{y}{5} < 1 \] Multiplying the entire inequality by 5, we find: \[ -5 < y < 5 \] Thus, the possible values of \( y \) are within the interval \( (-5, 5) \). Now, if \( y = 2 \), we can calculate the sum of the infinite geometric series \( S_{\infty} \). The formula for the sum of a convergent geometric series is given by: \[ S_{\infty} = \frac{a}{1 - r} \] where \( a \) is the first term, and \( r \) is the common ratio. Substituting \( y = 2 \): - The first term \( a = 5y = 5 \times 2 = 10 \). - The common ratio \( r = \frac{y}{5} = \frac{2}{5} \). Now we substitute these values into the sum formula: \[ S_{\infty} = \frac{10}{1 - \frac{2}{5}} = \frac{10}{\frac{3}{5}} = 10 \times \frac{5}{3} = \frac{50}{3} \] So, \( S_{\infty} = \frac{50}{3} \).