Final answer:
To find the first term and common ratio of the geometric series, use the formulas for the sum to infinity and the sum of the first two terms. Solve these two equations to find the values of a and r. Then, use the formula for the nth term to find the tenth term.
Step-by-step explanation:
To find the first term and common ratio of the geometric series, we can use the formulas for the sum to infinity and the sum of the first two terms. Let's call the first term 'a' and the common ratio 'r'.
The formula for the sum to infinity is: sum to infinity = a / (1 - r). We can substitute the given value of sum to infinity as 3/4 and solve for a / (1 - r) = 3/4.
The formula for the sum of the first two terms is: sum of first two terms = a + ar = 2/3. We can substitute the given value of sum of first two terms as 2/3 and solve for a + ar = 2/3.
Solving these two equations simultaneously, we can find the values of a and r. Once we have the values of a and r, we can use the formula for the nth term of a geometric series, which is a * r^(n-1), to find the tenth term.