Given the information that the fifth term of the geometric sequence is 781.25 and each term is [tex]\(\frac{1}{5}\)[/tex] of the following term, we need to determine which recursive formula represents this situation.
First, let's recall the general form of a geometric sequence's terms:
[tex]\[ a_n = a_1 \cdot r^{(n-1)}, \][/tex]
where [tex]\(a_1\)[/tex] is the first term, [tex]\(r\)[/tex] is the common ratio, and [tex]\(a_n\)[/tex] is the [tex]\(n\)[/tex]-th term.
In this problem, the fifth term [tex]\(a_5\)[/tex] is 781.25 and the common ratio [tex]\(r\)[/tex] is [tex]\(\frac{1}{5}\)[/tex].
### Step 1: Calculate the first term [tex]\(a_1\)[/tex]
Using the formula for the [tex]\(n\)[/tex]-th term of a geometric sequence:
[tex]\[ a_5 = a_1 \cdot \left(\frac{1}{5}\right)^{4}. \][/tex]
Given [tex]\(a_5 = 781.25\)[/tex]:
[tex]\[ 781.25 = a_1 \cdot \left(\frac{1}{5}\right)^{4}. \][/tex]
Solving for [tex]\(a_1\)[/tex]:
[tex]\[ 781.25 = a_1 \cdot \left(\frac{1}{625}\right), \][/tex]
[tex]\[ 781.25 = a_1 \cdot 0.0016, \][/tex]
[tex]\[ a_1 = \frac{781.25}{0.0016}. \][/tex]
[tex]\[ a_1 = 488281.25. \][/tex]
### Step 2: Identify the correct recursive formula
Now that we have the first term [tex]\(a_1 = 488281.25\)[/tex], we can check which of the given recursive formulas matches:
1. [tex]\(a_n = 5a_{n-1} ; a_1 = 1.25\)[/tex]
2. [tex]\(a_n = \frac{1}{5}a_{n-1} ; a_1 = 488281.25\)[/tex]
3. [tex]\(a_n = 5a_{n-1} ; a_1 = 488281.25\)[/tex]
4. [tex]\(a_n = \frac{1}{5}a_{n-1} ; a_1 = 1.25\)[/tex]
Given the first term [tex]\(a_1 = 488281.25\)[/tex] and the ratio [tex]\( \frac{1}{5} \)[/tex]:
- [tex]\(a_n = \frac{1}{5}a_{n-1} ; a_1 = 488281.25\)[/tex]
matches perfectly since:
- [tex]\(a_2 = \frac{1}{5} \cdot 488281.25 = 97656.25,\)[/tex]
- [tex]\(a_3 = \frac{1}{5} \cdot 97656.25 = 19531.25,\)[/tex]
- [tex]\(a_4 = \frac{1}{5} \cdot 19531.25 = 3906.25,\)[/tex]
- [tex]\(a_5 = \frac{1}{5} \cdot 3906.25 = 781.25.\)[/tex]
Thus, the correct recursive formula representing the situation is:
[tex]\[ a_n = \frac{1}{5} a_{n-1} ; a_1 = 488281.25. \][/tex]
Therefore, the answer is:
[tex]\[ \boxed{2} \][/tex]
