Shanna made an error in her formula [tex]\( f(x+1) = 2.5^{f(x)} \)[/tex]. Let's analyze the potential errors step-by-step:
1. Incorrect Common Ratio:
- This option implies that Shanna's formula is intended to represent a geometric sequence but she used the wrong common ratio.
- A geometric sequence has the form [tex]\( f(x+1) = f(x) \times r \)[/tex] where [tex]\( r \)[/tex] is the common ratio.
- Shanna's formula [tex]\( f(x+1) = 2.5^{f(x)} \)[/tex] does not resemble a geometric sequence form because it doesn't multiply [tex]\( f(x) \)[/tex] by a common ratio [tex]\( r \)[/tex].
2. Incorrect Initial Value:
- This option suggests that Shanna used the wrong initial value [tex]\( f(1) \)[/tex].
- However, the problem states that [tex]\( f(1) = 2 \)[/tex] and does not give any indication that this value is incorrect.
- Therefore, this option is not addressing her primary error.
3. Multiplying Instead of Using an Exponent:
- This option indicates that Shanna should have multiplied by [tex]\( f(x) \)[/tex] rather than using [tex]\( f(x) \)[/tex] as an exponent.
- However, there is no reason given in the problem that suggests multiplication is intended or correct in this context.
4. Treating the Sequence as Geometric Instead of Arithmetic:
- An arithmetic sequence has the form [tex]\( f(x+1) = f(x) + d \)[/tex] where [tex]\( d \)[/tex] is the common difference.
- Given that Shanna’s formula is [tex]\( f(x+1) = 2.5^{f(x)} \)[/tex], it is neither an arithmetic nor a geometric sequence.
- The error lies in that Shanna intended to describe a sequence but used an exponential function instead.
- Using [tex]\( 2.5^{f(x)} \)[/tex] suggests an exponential relationship rather than the algebraic relationship typical in arithmetic sequences.
Therefore, the correct answer is: She treated the sequence as geometric instead of arithmetic.
In summary, Shanna made the error of representing an exponential relationship rather than an arithmetic sequence in her formula.
