How many terms of each the following geometric series must be added to obtain the given sum? \( \begin{array}{ll}\text { (1) } 2+12+72+\ldots=18662 & \text { (2) } 8+\frac{8}{3}+\frac{8}{9}+\ldots=\frac{8744}{729} \\ \text { (3) }-4+16-64+\ldots=52428 & \text { (4) } \frac{3}{4}-\frac{3}{8}+\frac{3}{16}-\ldots=\frac{513}{1024}\end{array} \)

To determine the number of terms needed in each geometric series to achieve the given sum, we'll use the formula for the sum of the first \( n \) terms of a geometric series: \[ S_n = a \frac{r^n - 1}{r - 1} \] where: - \( a \) is the first term, - \( r \) is the common ratio, - \( S_n \) is the sum of the first \( n \) terms. Let's analyze each series individually. --- ### **(1) \( 2 + 12 + 72 + \ldots = 18,662 \)** - **First term (\( a \))**: 2 - **Common ratio (\( r \))**: \( \frac{12}{2} = 6 \) Using the sum formula: \[ 2 \frac{6^n - 1}{6 - 1} = 18,662 \\ \Rightarrow \frac{2(6^n - 1)}{5} = 18,662 \\ \Rightarrow 6^n - 1 = 46,656 \\ \Rightarrow 6^n = 46,656 \\ \Rightarrow 6^6 = 46,656 \] Thus, **\( n = 6 \)**. --- ### **(2) \( 8 + \frac{8}{3} + \frac{8}{9} + \ldots = \frac{8,744}{729} \)** - **First term (\( a \))**: 8 - **Common ratio (\( r \))**: \( \frac{8/3}{8} = \frac{1}{3} \) Using the sum formula: \[ 8 \frac{1 - (\frac{1}{3})^n}{1 - \frac{1}{3}} = \frac{8,744}{729} \\ \Rightarrow 12 \left(1 - \left(\frac{1}{3}\right)^n\right) = \frac{8,744}{729} \\ \Rightarrow \left(\frac{1}{3}\right)^n = \frac{1}{2,187} = \frac{1}{3^7} \\ \Rightarrow n = 7 \] Thus, **\( n = 7 \)**. --- ### **(3) \( -4 + 16 - 64 + \ldots = 52,428 \)** - **First term (\( a \))**: -4 - **Common ratio (\( r \))**: \( \frac{16}{-4} = -4 \) Using the sum formula: \[ -4 \frac{(-4)^n - 1}{-4 - 1} = 52,428 \\ \Rightarrow \frac{4((-4)^n - 1)}{5} = 52,428 \\ \Rightarrow (-4)^n = 65,536 \\ \Rightarrow 4^n = 65,536 \\ \Rightarrow 4^8 = 65,536 \\ \Rightarrow n = 8 \] Thus, **\( n = 8 \)**. --- ### **(4) \( \frac{3}{4} - \frac{3}{8} + \frac{3}{16} - \ldots = \frac{513}{1,024} \)** - **First term (\( a \))**: \( \frac{3}{4} \) - **Common ratio (\( r \))**: \( -\frac{1}{2} \) Using the sum formula: \[ \frac{3}{4} \frac{1 - \left(-\frac{1}{2}\right)^n}{1 - \left(-\frac{1}{2}\right)} = \frac{513}{1,024} \\ \Rightarrow \frac{1}{2} \left(1 - \left(-\frac{1}{2}\right)^n\right) = \frac{513}{1,024} \\ \Rightarrow \left(-\frac{1}{2}\right)^n = -\frac{1}{512} \\ \Rightarrow n = 9 \] Thus, **\( n = 9 \)**. --- ### **Summary of Results:** 1. **6 terms** 2. **7 terms** 3. **8 terms** 4. **9 terms** **Answer Summary:** Problem (1) requires 6 terms; (2) 7 terms; (3) 8 terms; and (4) 9 terms must be added.