Let's clarify the situation with the mixed number representation and the options provided.
We found that the 9th term \( A_9 \) is \( \frac{65}{4} \).
Converting \( \frac{65}{4} \) to a mixed number yields \( 16 \frac{1}{4} \), but this seems outside the options you've provided.
Since the options you have are:
- \( 4 \frac{1}{4} \)
- \( 8 \frac{1}{4} \)
- \( 8 \frac{3}{4} \)
- \( 4 \frac{3}{4} \)
It appears there may be a misunderstanding regarding the options based on either another term calculation or the sequence definition itself.
Re-checking the Calculation
To be completely accurate:
-
We have the explicit formula:
\[
A_n = \frac{1}{4} + (n-1) \cdot 2
\]
-
For \( n = 9 \):
\[
A_9 = \frac{1}{4} + (9-1) \cdot 2 = \frac{1}{4} + 8 \cdot 2 = \frac{1}{4} + 16 = \frac{1 + 64}{4} = \frac{65}{4}
\]
This gives us \( 16 \frac{1}{4} \) as confirmed.
If we were looking for the 4th term instead:
Calculating the 4th Term
\[
A_4 = \frac{1}{4} + (4-1) \cdot 2 = \frac{1}{4} + 3 \cdot 2 = \frac{1}{4} + 6 = \frac{1 + 24}{4} = \frac{25}{4} = 6 \frac{1}{4}
\]
This is also not in your options.
Verifying Other Terms
If you want to see how it connects:
- \( A_1 = \frac{1}{4} \)
- \( A_2 = \frac{1}{4} + 2 = 2 \frac{1}{4} \)
- \( A_3 = 2 \frac{1}{4} + 2 = 4 \frac{1}{4} \)
- \( A_4 = 6 \frac{1}{4} \) (not included in options)
Let me clarify:
You can only provide options 4 \( \frac{1}{4} \), 8 \( \frac{1}{4} \), 8 \( \frac{3}{4} \) or 4 \( \frac{3}{4} \). The original formula yields \(16.25\) or \(16 \frac{1}{4}\), which means if you made a mistake on the term number or if there was a check on the values being sought.
Final Verifications
Can you please confirm the \(n\) term you wish to evaluate? It may be that you need a different \(n\). The options \(4 \frac{1}{4}\), \(8 \frac{1}{4}\), \(8 \frac{3}{4}\), or \(4 \frac{3}{4}\) don’t match \(n = 9\).
