Which of the following statements is/are true about the series the summation from n=0 to infinity of (3/2)^n?
I. It diverges.
II. It is geometric.
III. It is arithmetic
Possible answers:
a) I only
b) II only
c) I and II only
d) I and III only
To determine which statements are true about the series, let's analyze each option:
I. The series diverges.
To determine if a series diverges, we need to check if the sequence of partial sums grows infinitely. In this case, the series is (3/2)^n, which is a geometric series with a common ratio of 3/2. Geometric series diverge if the absolute value of the common ratio is greater than or equal to 1. Since the absolute value of 3/2 is greater than 1, the series diverges.
II. The series is geometric.
A geometric series is one in which each term can be obtained by multiplying the previous term by a fixed, non-zero constant called the common ratio. In this case, the series (3/2)^n can indeed be obtained by multiplying the previous term by the constant 3/2. Therefore, the series is geometric.
III. The series is arithmetic.
An arithmetic series is one in which each term can be obtained by adding a fixed, non-zero constant called the common difference to the previous term. However, in this case, the series (3/2)^n does not meet this condition as the terms are not obtained by adding a common difference. Therefore, the series is not arithmetic.
Based on the analysis above:
a) I only: Statement I is true because the series diverges.
b) II only: Statement II is true because the series is geometric.
c) I and II only: Both statements I and II are true.
d) I and III only: Statement III is false because the series is not arithmetic.
Therefore, the correct answer is (c) I and II only.