Shanna made an error in her treatment of the sequence. Specifically, she treated the sequence as geometric instead of arithmetic. Here’s a detailed explanation of the mistake:
### Explanation of the Error
1. Geometric vs. Arithmetic Sequence:
- Geometric Sequence: A sequence where each term is found by multiplying the previous term by a constant ratio. The general form is [tex]\( a_n = a_1 \cdot r^{(n-1)} \)[/tex], where [tex]\( r \)[/tex] is the common ratio.
- Arithmetic Sequence: A sequence where each term is found by adding a constant difference to the previous term. The general form is [tex]\( a_n = a_1 + (n-1)d \)[/tex], where [tex]\( d \)[/tex] is the common difference.
2. Shanna’s Formula:
- Shanna used the formula [tex]\( f(x+1) = 2.5^{f(x)} \)[/tex] with the initial value [tex]\( f(1) = 2 \)[/tex].
3. Analysis of Shanna’s Formula:
- The given formula [tex]\( f(x+1) = 2.5^{f(x)} \)[/tex] implies that each term [tex]\( f(x) \)[/tex] depends on the exponentiation of the previous term, suggesting a relationship more complex than simple multiplication or addition.
- This formula suggests an exponential function, which is neither a geometric sequence (which involves multiplication by a fixed ratio) nor an arithmetic sequence (which involves addition of a fixed difference).
4. Correct Understanding for Arithmetic Sequence:
- For an arithmetic sequence, the relationship should involve a fixed difference [tex]\( d \)[/tex].
- The formula for the next term in an arithmetic sequence would be [tex]\( f(x+1) = f(x) + d \)[/tex], indicating that the next term is a constant increment more than the previous term.
### Conclusion
Therefore, Shanna’s error was treating the sequence as having a geometric nature when it should have been treated as arithmetic, where each term increases by a fixed amount rather than by a exponential factor.
