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!

Question

UpStudy AI · Accepted Answer

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.**
Both Andre's and Lin's equations are correct because they both produce the same sequence of values starting from $$ n = 1 $$.

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