Problem 51 - College Algebra

Question

Graph the first five terms of the indicated sequence. $$a_{n}=\frac{(n+1) !}{(n-1) !}$$

Step-by-Step Solution

Answer

Plot the points (1, 2), (2, 6), (3, 12), (4, 20), and (5, 30) on a graph.

Step 1: Understanding the Sequence Formula

The given sequence is defined by the formula \(a_{n}=\frac{(n+1) !}{(n-1) !}\). This involves factorials, so we'll need to compute the factorial of \((n+1)\) and \((n-1)\) for the first few values of \(n\).

Step 2: Calculate \(a_{1}\)

For \(n=1\), substitute into the formula: \(a_{1} = \frac{(1+1)!}{(1-1)!} = \frac{2!}{0!} = \frac{2}{1} = 2\).

Step 3: Calculate \(a_{2}\)

For \(n=2\), substitute into the formula: \(a_{2} = \frac{(2+1)!}{(2-1)!} = \frac{3!}{1!} = \frac{6}{1} = 6\).

Step 4: Calculate \(a_{3}\)

For \(n=3\), substitute into the formula: \(a_{3} = \frac{(3+1)!}{(3-1)!} = \frac{4!}{2!} = \frac{24}{2} = 12\).

Step 5: Calculate \(a_{4}\)

For \(n=4\), substitute into the formula: \(a_{4} = \frac{(4+1)!}{(4-1)!} = \frac{5!}{3!} = \frac{120}{6} = 20\).

Step 6: Calculate \(a_{5}\)

For \(n=5\), substitute into the formula: \(a_{5} = \frac{(5+1)!}{(5-1)!} = \frac{6!}{4!} = \frac{720}{24} = 30\).

Step 7: Graph the Sequence Terms

Now that we have the values \(a_{1} = 2\), \(a_{2} = 6\), \(a_{3} = 12\), \(a_{4} = 20\), and \(a_{5} = 30\), plot these points on a graph with \(n\) on the x-axis and \(a_n\) on the y-axis to visualize the sequence.

Key Concepts

Factorials

Factorials are an important mathematical concept used throughout our sequence calculations. The factorial of a positive integer \(n\), denoted as \(n!\), is the product of all positive integers less than or equal to \(n\). For instance, \(3! = 3 \times 2 \times 1 = 6\). The factorial grows very quickly as \(n\) increases.

• One of the key properties of factorials is that \(0! = 1\).
• Factorials are instrumental in permutations and combinations, which help solve problems involving arrangements and selections.

In the context of the sequence \(a_{n} = \frac{(n+1)!}{(n-1)!}\), we compute the factorial of \((n+1)\) and \((n-1)\) separately for the first few terms. Understanding factorials aids in simplifying such expressions and analyzing how sequences behave.

Sequence Visualization

Sequence visualization helps us understand the behavior of a sequence by seeing its values represented on a coordinate plane. For instance, by calculating the first few terms of a sequence and plotting them, we gain insight into the pattern and direction that the sequence is following.

• Visualizing sequences can reveal whether they are increasing, decreasing, or converging toward a particular value.
• It allows for a better intuition about the sequence beyond numerical calculations.

In our exercise, we calculated:

• \(a_1 = 2\)
• \(a_2 = 6\)
• \(a_3 = 12\)
• \(a_4 = 20\)
• \(a_5 = 30\)

These values show a rapidly increasing pattern as \(n\) increases. This visualization can direct further exploration or hypotheses about the sequence's behavior as \(n\) continues.

Graphical Representation

The graphical representation of sequences involves plotting the calculated terms on a graph, which provides a clear and immediate picture of how the sequence behaves over time. By representing \(n\) on the x-axis and \(a_n\) on the y-axis, we can grasp the relationship between these two variables easily.

• Graphing a sequence makes trends and patterns apparent, such as linear, exponential, or quadratic growth.
• It simplifies comparison with other sequences or functions by providing a visual context.

In our example, plotting the points \((1, 2), (2, 6), (3, 12), (4, 20), (5, 30)\) creates a graph that visually demonstrates the significant increase in values as \(n\) increases. This strong visual tool shows that the sequence values do not only get larger; they increase at a more accelerated rate as \(n\) continues past what we've initially analyzed.