Final answer:
The first term of the geometric series is 12/7. Since the common ratio is greater than 1, the series diverges, and thus, there is no finite sum to infinity (S_infinity). Accordingly, it is impossible to find a value of n for which the infinite sum minus the sum of the first n terms is less than 0.001.
Step-by-step explanation:
The student is tasked with finding the first term and sum to infinity of a geometric sequence, as well as the least number of terms required for the sum to approximate to infinity within a certain error margin.
Finding the first term (u_1)
The first term, u_1, of the geometric sequence is given by the general formula for the sum of a geometric series when r=1. Since the sum is S_n = \sum_{r=1}^{n} (\frac{3}{2} \times (\frac{8}{7})^r), the first term u_1 is simply when r=1, which is u_1 = \frac{3}{2} \times (\frac{8}{7})^1 = \frac{3}{2} \times \frac{8}{7} = \frac{12}{7}.
Finding the sum to infinity (S_infinity)
For the sum to infinity of a geometric sequence, S_infinity, we use the formula S_infinity = \frac{u_1}{1 - r} where r is the common ratio (\frac{8}{7}). Since |r| > 1, the sum to infinity does not exist because the terms increase without bound.
Least value of n for S_infinity - S_n < 0.001
For the least value of n such that S_infinity - S_n < 0.001, this cannot be determined as S_infinity is not finite in this case because the common ratio is greater than 1.