Final answer:
To solve the problem, we use the formula for the sum of a finite geometric series and set up a system of equations. We can then solve the system to find the values of the first term and the common ratio. Finally, we use these values to determine the first three terms and find their product, which is -1/5.
Step-by-step explanation:
To solve this problem, we can use the formula for the sum of a finite geometric series:
S = a(1 - r^n)/(1 - r)
where S is the sum, a is the first term, r is the common ratio, and n is the number of terms. We are given that the sum of the first three terms is -7/10, so we can set up the equation:
-7/10 = a(1 - r^3)/(1 - r)
We are also given that the product of the first three terms is -1/125, so we can set up another equation:
-1/125 = a^3 r^3
Now we have a system of equations that we can solve to find the values of a and r. Once we have a and r, we can use them to find the first three terms: a/r, a, and ar.
To get to -1/5, you need to find the product of these three terms. Substitute the values of a/r, a, and ar into the equation a(1 - r^n)/(1 - r) and simplify to find the sum of the first three terms. Then, find the product of these three terms, which will give you -1/5.