Question

Determine whether the sequence with th given \(n\) th term is monotonic. Discuss the boundedness of th sequence. Use a graphing utility to confirm your results. \(a_{n}=\left(\frac{2}{3}\right)^{n}\)

Short answer

The sequence \(a_{n}=\left(\frac{2}{3}\right)^{n}\) is decreasing (monotonic) and semi-bounded, being bounded from below by zero.

Step-by-step solution

Determine the nature of the sequence

First, the given sequence is \(a_{n}=\left(\frac{2}{3}\right)^{n}\) and it's clear from the form of the sequence that it diminishes as n increases. Each term is \(2/3\) of the previous term, meaning the series decreases with each subsequent term. Therefore, the sequence is decreasing and hence, monotonic.

Determine boundedness

Next, despite being decreasing, the sequence never decreases below zero since the terms of the sequence are positive for all natural numbers. This means that the sequence has a lower bound, which is 0. However, there isn't an upper bound for the sequence since it starts at a finite value (\(a_{1}=2/3\) when n = 1), but decreases indefinitely. Hence, the sequence is bounded from below, but unbounded from above. So, we say the sequence is semi-bounded.

Use graphing utility for confirmation

Plot the sequence using a graphing utility. The x-axis should represent n, and the y-axis should represent \(a_{n}\). Visually, it can be observed that the sequence decreases as n increases, and stays above the x-axis, which confirms the sequence is decreasing and is bounded below by zero.