Unit 1 - Number and Algebra
Required Prior Knowledge
Questions
Simplify
a) \(\frac{3x^{2}+6x}{4x} \)
b) \(\frac{x^{2}-5x+6}{x^{2}+x-6}\)
Solutions
Get Ready
Questions
There are 4 people in a room.
They all shake hands with the other 3 people. How many handshakes take place?
The 4 people now line up to leave the room. How many different ways are there for them to line up?
Solutions
Notes
\(n!\) is read as “\(n\) factorial” and$$n!=n\times\left(n-1\right)\times\left(n-2\right)\times\left(n-3\right)\times\dots\times 2\times 1$$e.g. \(5!=5\times 4\times 3\times 2\times 1=120\).
One of the most useful properties of factorials is that$$n!=n\times\left(n-1\right)!$$e.g. \(5!=5\times 4!\).
By definition, \(0!=1\).
Below is a joke that makes good use of factorials.
Solve \(230-220\times 0.5\).
You might not believe me, but the answer is 5!
Examples and Your Turns
Example
Evaluate \(4!\).
Evaluate$$\frac{10!\times 5!}{7!\times 6!}$$
Evaluate$$\frac{7!}{4!\times 3!}$$
Evaluate $$\frac{5!}{3!}$$
Example
Express as a ratio of factorials$$10\times 9\times 8\times 7$$
Your Turn
Express as a ratio of factorials$$12\times 11\times 10$$
Example
Express as a ratio of factorials$$\frac{10\times 9\times 8\times 7}{4\times 3\times 2\times 1}$$
Your Turn
Express as a ratio of factorials$$\frac{11\times 10\times 9\times 8\times 7}{5\times 4\times 3\times 2\times 1}$$
Example
Simplify \(9!-7!\) by factorising
Your Turn
Simplify \(10!-9!+8!\) by factorising
Example
Simplify$$\frac{\left(n+1\right)!}{\left(n-1\right)!}$$
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$$\begin{align}\frac{\left(n+1\right)!}{\left(n-1\right)!}&=\frac{\left(n+1\right)n\left(n-1\right)!}{\left(n-1\right)!}\\&=n\left(n+1\right)\end{align}$$
Your Turn
Simplify$$\frac{\left(n+1\right)!+n!}{\left(n-1\right)!}$$
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$$\begin{align}\frac{\left(n+1\right)!+n!}{\left(n-1\right)!}&=\frac{\left(n+1\right)n\left(n-1\right)!+n\left(n-1\right)!}{\left(n-1\right)!}\\&=\frac{\left(n-1\right)!\left(n\left(n+1\right)+n\right)}{\left(n-1\right)!}\\&=n\left(n+1\right)+n\\&=n^{2}+2n\end{align}$$
Your Turn
Find the value of \(x\in\mathbb{R}\), given that$$\frac{\left(x^{2}-1\right)!}{x^{2}-1}=23!$$
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$$\begin{align}\frac{\left(x^{2}-1\right)!}{\left(x^{2}-1\right)!}&=\frac{\left(x^{2}-1\right)\left(x^{2}-2\right)!}{\left(x^{2}-1\right)!}\\&=\left(x^{2}-2\right)!\end{align}$$Since this is equal to \(23!\) we have that$$\left(x^{2}-2\right)!=23!\\ \therefore x^{2}-2=23 \\ x^{2}=25 \\ x=\pm 5$$
Your Turn
Show that there are exactly:
a) \(6!\) minutes in \(12\) hours
b) \(10!\) seconds in \(6\) weeks
c) \(8!\) [time units] in \(n\) weeks, where the time unit and value of \(n\) are to be found.
Your Turn
Prove by induction that for \(n\in\mathbb{Z}^{+}\),$$\sum_{r=1}^{n} \frac{r\times 2^{r}}{\left(r+2\right)!}=1-\frac{2^{n+1}}{\left(n+2\right)!}$$
Your Turn
Prove by induction that for \(n\in\mathbb{Z}^{+}\),$$\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\dots+\frac{n}{\left(n+1\right)!}=1-\frac{1}{\left(n+1\right)!}$$
Your Turn
Prove by induction that for \(n\in\mathbb{Z}^{+}\),$$\sum_{i=1}^{n} \left(i^{2}+1\right)i!=n\left(n+1\right)!$$
Your Turn
Prove by induction that for \(n\in\mathbb{Z}^{+},n\ge 2\),$$\left(2n\right)!>2^{n}\left(n!\right)^{2}$$
Investigation
a) Calculate the value of$$\frac{1}{0!}$$
b) Calculate the value of$$\frac{1}{0!}+\frac{1}{1!}$$
c) Calculate the value of$$\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}$$
d) Calculate the value of$$\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}$$
e) Calculate the value of$$\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\dots +\frac{1}{8!}$$
f) Write the series \(\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\dots +\frac{1}{8!}\) in sigma notation
g) What do you notice about the value of these series as the number of terms increases?
h) Use sigma notation to evaluate the following using your graphical calculator$$\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\dots +\frac{1}{20!}$$
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.