2. Andre and Lin are asked to write an equation for \( S \) that isn't recursive. Andre writes \( S(n)=3^{n} \) for \( n \geq 0 \) while Lin writes \( S(n)=3^{n-1} \) for \( n \geq 1 \). Whose equation do you think is correct? A I think Andre's equation is correct. B I think Lin's equation is correct. C I think both equations are correct. D I think neither is correct. Explain your reasoning!

To determine whose equation for \( S \) is correct, let's analyze both equations provided by Andre and Lin. 1. **Andre's Equation:** \[ S(n) = 3^n \quad \text{for } n \geq 0 \] - This equation starts at \( n = 0 \). - For \( n = 0 \), \( S(0) = 3^0 = 1 \). - For \( n = 1 \), \( S(1) = 3^1 = 3 \). - For \( n = 2 \), \( S(2) = 3^2 = 9 \). - The values of \( S(n) \) are \( 1, 3, 9, 27, \ldots \) which is a geometric sequence with a common ratio of 3. 2. **Lin's Equation:** \[ S(n) = 3^{n-1} \quad \text{for } n \geq 1 \] - This equation starts at \( n = 1 \). - For \( n = 1 \), \( S(1) = 3^{1-1} = 3^0 = 1 \). - For \( n = 2 \), \( S(2) = 3^{2-1} = 3^1 = 3 \). - For \( n = 3 \), \( S(3) = 3^{3-1} = 3^2 = 9 \). - The values of \( S(n) \) are \( 1, 3, 9, 27, \ldots \) which is also a geometric sequence with a common ratio of 3. ### Comparison: - Both equations yield the same sequence of values starting from \( n = 1 \) for Lin's equation and from \( n = 0 \) for Andre's equation. - Andre's equation provides a value for \( n = 0 \), while Lin's equation does not define \( S(0) \). ### Conclusion: - Since both equations produce the same values for \( n \geq 1 \) and Andre's equation includes \( n = 0 \) with a defined value, we can conclude that both equations are correct, but Andre's is more comprehensive as it covers the case for \( n = 0 \). Thus, the correct answer is: **C I think both equations are correct.**