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. \(a_{n}=\left(\frac{3}{2}\right)^{n}\)

Short answer

The sequence \(a_{n}=\left(\frac{3}{2}\right)^{n}\) is an increasing (monotonic) sequence and it is bounded below by 0. However, it is unbounded above.

Step-by-step solution

Identifying the nature of the sequence

An important observation for this sequence \(a_{n}=\left(\frac{3}{2}\right)^{n}\) is that it is a geometric sequence, where the common ratio \(r = \frac{3}{2}\) is positive and greater than 1. As each term of the sequence is multiplied by a common ratio greater than 1, the sequence increases by a larger amount each term. Hence, the sequence is increasing, which indicates that the sequence is monotonic.

Testing for boundedness

For testing boundedness, one needs to determine if there is any upper or lower limit for the sequence. Since this sequence keeps increasing due to the positive common ratio, the sequence is unbounded from above. However, this sequence is greater than 0 for all \(n\) because \(\left(\frac{3}{2}\right)^{n} > 0\) for all \(n\). Thus, it is bounded below by 0.

Confirming with graphic analysis

To confirm the above findings, plot the function \(y=\left(\frac{3}{2}\right)^{x}\) for a range of \(x\) values. The graph will reveal an increasing curve starting at \(y>0\), which supports the statement of the sequence being increasing and bounded below.

Key concepts

Geometric Sequence

A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a fixed non-zero number called the common ratio. This common ratio is a key characteristic that defines the growth or decay of the sequence. For example, in the sequence given by the formula \(a_{n} = \left(\frac{3}{2}\right)^{n}\), the common ratio is \(\frac{3}{2}\), which is greater than 1. This means that each successive term is 1.5 times larger than the one before it.

Visualizing this, if the first term is 1, the next terms would be 1.5, 2.25, 3.375, and so on, clearly showing a pattern of increase. In educational terms, the geometric sequence lets us categorize patterns of growth effectively, making it essential for understanding exponential phenomena in various fields like finance, biology, and physics. When studying such sequences, always note the initial term and the common ratio, as they are vital for determining its properties including monotonicity and boundedness.

Boundedness in Sequences

When we discuss boundedness in the context of sequences, we're essentially talking about limits—whether there are any constraints on how high or low the values of the sequence can go. A sequence is said to be bounded above if there is a real number that is greater than or equal to every term of the sequence. Conversely, it is bounded below if there is a real number that is less than or equal to every term.

A sequence that is both bounded above and below is called simply 'bounded'. For example, the sequence \(\left(\frac{3}{2}\right)^{n}\) is bounded below by 0 because no term in the sequence can be negative (as a positive number raised to any power remains positive). However, this sequence is not bounded above because there is no real number that can cap the growth—each term multiplies by 1.5, growing larger indefinitely. Understanding boundedness is essential for determining the long-term behavior of sequences, which is particularly useful in calculus and analysis when dealing with infinite series and convergence.

Graphing Sequences

Graphing sequences allows us to visually inspect the behavior and properties of sequences, which is particularly useful when it comes to understanding monotonicity and boundedness. To graph the sequence \(a_{n} = \left(\frac{3}{2}\right)^{n}\), we consider discrete points where each term of the sequence corresponds to a point on the graph, with \(n\) as the x-axis (independent variable) and \(a_{n}\) as the y-axis (dependent variable).

As we plot the points for various values of \(n\), we will observe a curve that rises to the top right of the graph. This curve would not intersect with any horizontal line drawn above the x-axis, confirming the sequence's unbounded nature from above. At the same time, it will sit above the x-axis, illustrating that it is bounded below by 0. Graphical representations like this are an effective aid in getting a more intuitive understanding of sequences for educational purposes and can be a supportive tool to confirm algebraic findings.