Question

Factorial function The factorial function is defined for positive integers as \(n !=n(n-1)(n-2) \cdots 3 \cdot 2 \cdot 1\) a. Make a table of the factorial function, for \(n=1,2,3,4,5\) b. Graph these data points and then connect them with a smooth curve. c. What is the least value of \(n\) for which \(n !>10^{6} ?\)

Short answer

Answer: The least value of n for which n! > 10^6 is n = 10.

Step-by-step solution

(Step 1: Calculate Factorials)

To calculate the factorial function for \(n=1,2,3,4,5\), we will compute the product of all positive integers up to \(n\). For \(n=1\): \(1! = 1\) For \(n=2\): \(2! = 2 \cdot 1 = 2\) For \(n=3\): \(3! = 3 \cdot 2 \cdot 1 = 6\) For \(n=4\): \(4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24\) For \(n=5\): \(5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120\)

(Step 2: Create the Table)

Using the calculations from Step 1, we will make a table of the factorial function for \(n=1,2,3,4,5\): | n | Factorial (n!) | |----|---------------| | 1 | 1 | | 2 | 2 | | 3 | 6 | | 4 | 24 | | 5 | 120 |

(Step 3: Graph the Data Points)

Using the table from Step 2, we will graph the data points on a two-dimensional plane. The x-axis represents the values of \(n\), and the y-axis represents the corresponding factorials (n!). Plot the points \((1,1)\), \((2,2)\), \((3,6)\), \((4,24)\), and \((5,120)\) on the graph. Then, connect them with a smooth curve to illustrate the growth of the factorial function.

(Step 4: Find the Least Value of n for which \(n!>10^{6}\))

We need to find the smallest value of \(n\) such that \(n! > 10^6\). We can do this by calculating factorials incrementally and comparing the result to \(10^6\). We already have the values for \(n = 1, 2, 3, 4, 5\), so we can start with \(n = 6\). For \(n=6\): \(6! = 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 720\) For \(n=7\): \(7! = 7 \cdot 6! = 7\cdot 720 = 5040\) For \(n=8\): \(8! = 8 \cdot 7! = 8 \cdot 5040 = 40320\) For \(n=9\): \(9! = 9 \cdot 8! = 9 \cdot 40320 = 362880\) For \(n=10\): \(10! = 10 \cdot 9! = 10\cdot 362880 = 3628800\) We can see that \(10! > 10^6\). Therefore, the least value of \(n\) for which \(n! > 10^6\) is \(n = 10\).

Key concepts

Calculating Factorials

Understanding how to calculate factorials is foundational in various branches of mathematics, including combinatorics and probability. The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. It's formally defined only for non-negative integers.

For example, when calculating the factorial of 5, represented as 5!, it involves multiplying all positive integers from 1 up to 5 together:
\[5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\].

While the concept might seem simple at first, factorials grow very rapidly with each additional number, so much so that already at 5!, we reach a product of 120. This exponential increase makes calculating larger factorials by direct multiplication impractical and instead requires more sophisticated computational methods or factorial functions on calculators.

Graphing Factorials

Graphing factorial values gives us a visual representation of their rapid growth. To graph the factorial function, we can plot the values of the function on a coordinate plane with the input number n along the x-axis and the factorial of n (n!) along the y-axis.

As an exercise, let's consider plotting data points for n = 1 through n = 5. The coordinate pairs would be (1, 1!), (2, 2!), (3, 3!), (4, 4!), and (5, 5!). Once plotted, the points are generally connected with a smooth curve to convey the continuous nature of the factorial function's growth, even though the function is only defined for integer values of n.

The result is a steep, upward-sloping curve that reveals the function's rapid increase, which becomes particularly dramatic as the value of n increases. This type of visualization helps students grasp the concept of factorial growth apart from the numerical values.

Growth of Factorial Function

The graphing exercise from the previous section clearly demonstrates the growth pattern of the factorial function: it is highly exponential. This means with each incremental increase in n, the factorial n! increases at a much faster rate than polynomial functions such as n^2 or n^3.

This rapid growth has significant implications in mathematics, particularly in permutations and combinations where factorial calculations help determine the number of ways in which events can occur. However, beyond certain values, this rapid growth makes it almost impossible to compute factorials through regular means due to limitations in computer processing and storage capabilities.

For instance, the exercise of finding the smallest integer n such that n! > 10^6 shows an important characteristic of the factorial function: not only does n! increase quickly, but each additional multiplication has a compounding effect that predicates a need for logarithmic or approximative methods for large n in practical applications.