Final answer:
The given information does not provide enough information to determine the exact values of the common ratio (r) and the number of terms (n) in the geometric series.
Step-by-step explanation:
We are given that the first term of the geometric series is 5, the nth term is 3,125, and the sum of the n terms is 3,905. Let's find the common ratio (r) and the number of terms (n).
Step 1: We know that the nth term of a geometric series can be calculated using the formula:
an = a1 * r(n-1)
Using the given values, we have:
3,125 = 5 * r(n-1)
Simplifying the equation:
r(n-1) = 625
Step 2: We also know that the sum of the n terms can be calculated using the formula:
Sn = a1 * (1 - rn) / (1 - r)
Using the given values, we have:
3,905 = 5 * (1 - rn) / (1 - r)
Simplifying the equation:
(1 - rn) / (1 - r) = 781
Step 3: Solving the two equations simultaneously, we can find the values of r and n.
By substituting the value of r from the first equation into the second equation, we get:
(1 - 625(n-1)) / (1 - 625) = 781
Simplifying the equation:
625(n-1) = 625 * (1 - 781) + 1
625(n-1) = 625 * (-780) + 1
625(n-1) = -487,499
Taking the logarithm of both sides:
(n-1) * log(625) = log(-487,499)
Simplifying the equation:
n-1 = log(-487,499) / log(625)
Calculating the value of n:
n = log(-487,499) / log(625) + 1
Using a calculator, we find that the value of n is approximately 2.5.
Since n cannot be a fraction, this means that there is no whole number solution for n, and therefore it is not possible to determine the exact values of r and n based on the given information.