Consider the geometric series: \( 45+15+5+\ldots \) 3.1 Does the sequence converge or diverge? Justify your answe 3.2 Determine the sum to infinity of the series, if it exists.

Let's analyze the given geometric series: \[ 45 + 15 + 5 + \ldots \] ### 3.1 Convergence or Divergence A geometric series converges if the absolute value of the common ratio \( |r| \) is less than 1. To determine the common ratio: \[ r = \frac{\text{second term}}{\text{first term}} = \frac{15}{45} = \frac{1}{3} \] Since: \[ |r| = \left|\frac{1}{3}\right| = \frac{1}{3} < 1 \] **Conclusion:** The series **converges** because the common ratio \( |r| \) is less than 1. ### 3.2 Sum to Infinity For a convergent geometric series, the sum to infinity \( S_\infty \) is given by: \[ S_\infty = \frac{a}{1 - r} \] Where: - \( a = 45 \) (the first term) - \( r = \frac{1}{3} \) (the common ratio) Plugging in the values: \[ S_\infty = \frac{45}{1 - \frac{1}{3}} = \frac{45}{\frac{2}{3}} = 45 \times \frac{3}{2} = \frac{135}{2} = 67.5 \] **Conclusion:** The sum to infinity of the series is **67.5**. Answer: 3.1 The series converges since its common ratio |1⁄3| is less than 1; 3.2 Its sum to infinity is 67.5.