1 Determine the sum to infinity for each of these geometric series: \( 1.136+12+4+\ldots \) \( 1.2-20-5-1,25-\ldots \) \( 1.3 \quad 40-20+10-5+\ldots \) \( 1.4 \quad-27+18-12+8-\ldots \)

To determine the sum to infinity of each geometric series, we'll use the formula for the sum of an infinite geometric series: \[ S = \frac{a_1}{1 - r} \] where: - \( S \) is the sum to infinity, - \( a_1 \) is the first term of the series, - \( r \) is the common ratio between consecutive terms. **Note:** This formula is valid only if the absolute value of the common ratio satisfies \( |r| < 1 \). Let's analyze each series one by one. --- ### **1.1. \( 36 + 12 + 4 + \ldots \)** - **First Term (\( a_1 \))**: 36 - **Second Term**: 12 **Common Ratio (\( r \))**: \[ r = \frac{\text{Second Term}}{\text{First Term}} = \frac{12}{36} = \frac{1}{3} \] **Verification of Convergence**: \[ |r| = \left| \frac{1}{3} \right| < 1 \quad \text{(Converges)} \] **Sum to Infinity (\( S \))**: \[ S = \frac{36}{1 - \frac{1}{3}} = \frac{36}{\frac{2}{3}} = 36 \times \frac{3}{2} = 54 \] --- ### **1.2. \( -20 - 5 - 1.25 - \ldots \)** - **First Term (\( a_1 \))**: -20 - **Second Term**: -5 **Common Ratio (\( r \))**: \[ r = \frac{\text{Second Term}}{\text{First Term}} = \frac{-5}{-20} = 0.25 \] **Verification of Convergence**: \[ |r| = |0.25| < 1 \quad \text{(Converges)} \] **Sum to Infinity (\( S \))**: \[ S = \frac{-20}{1 - 0.25} = \frac{-20}{0.75} = -\frac{80}{3} \approx -26.67 \] --- ### **1.3. \( 40 - 20 + 10 - 5 + \ldots \)** - **First Term (\( a_1 \))**: 40 - **Second Term**: -20 **Common Ratio (\( r \))**: \[ r = \frac{\text{Second Term}}{\text{First Term}} = \frac{-20}{40} = -0.5 \] **Verification of Convergence**: \[ |r| = | -0.5 | = 0.5 < 1 \quad \text{(Converges)} \] **Sum to Infinity (\( S \))**: \[ S = \frac{40}{1 - (-0.5)} = \frac{40}{1 + 0.5} = \frac{40}{1.5} = \frac{80}{3} \approx 26.67 \] --- ### **1.4. \( -27 + 18 - 12 + 8 - \ldots \)** - **First Term (\( a_1 \))**: -27 - **Second Term**: 18 **Common Ratio (\( r \))**: \[ r = \frac{\text{Second Term}}{\text{First Term}} = \frac{18}{-27} = -\frac{2}{3} \] **Verification of Convergence**: \[ |r| = \left| -\frac{2}{3} \right| = \frac{2}{3} < 1 \quad \text{(Converges)} \] **Sum to Infinity (\( S \))**: \[ S = \frac{-27}{1 - (-\frac{2}{3})} = \frac{-27}{1 + \frac{2}{3}} = \frac{-27}{\frac{5}{3}} = -27 \times \frac{3}{5} = -\frac{81}{5} = -16.2 \] --- ### **Summary of Sums to Infinity** 1. **\( 36 + 12 + 4 + \ldots \)**: \( S = 54 \) 2. **\( -20 - 5 - 1.25 - \ldots \)**: \( S = -\frac{80}{3} \) or approximately -26.67 3. **\( 40 - 20 + 10 - 5 + \ldots \)**: \( S = \frac{80}{3} \) or approximately 26.67 4. **\( -27 + 18 - 12 + 8 - \ldots \)**: \( S = -\frac{81}{5} \) or -16.2 --- Each of these sums is based on identifying the first term and the common ratio, ensuring the series converges (\( |r| < 1 \)), and then applying the sum to infinity formula for geometric series.