Unit 1 - Number and Algebra
Required Prior Knowledge
Questions
Solve this pair of simultaneous equations.
$$\begin{matrix}5x & + & 9y & = & -14 \\ 2x & + & 6y & = & 4 \end{matrix}$$
Solutions
Get Ready
Questions
What is the same and what is different about these three sequences?
$$3, 8, 13, 18, 23, …$$
$$u_{1}=3, u_{n+1}=u_{n}+8$$
$$a_{n}=4n-1$$
Solutions
Notes
An arithmetic sequence is a sequence where successive terms differ by the same amount.
This difference between terms is known as the common difference and is usually denoted with \(d\).
A sequence is arithmetic if and only if \(u_{n+1}-u_{n}=d\) for all \(n\in\mathbb{Z}^{+}\).
That is, if $$u_{2}-u_{1}=u_{3}-u_{2}=u_{4}-u_{3}=d$$
Consider the arithmetic sequence with \(u_{1}=3\) and \(d=2\).
Work out:
\(u_{2}\)
\(u_{3}\)
\(u_{4}\)
\(u_{5}\)
\(u_{n}\)
-
\(u_{2}=3+2=5\)
\(u_{3}=5+2=7\)
\(u_{4}=7+2=9\)
\(u_{5}=9+2=11\)
\(u_{n}=u_{n-1}+2=2n+1\)
Now consider the arithmetic sequence with \(u_{1}=u_{1}\) and \(d=d\).
Work out:
\(u_{2}\)
\(u_{3}\)
\(u_{4}\)
\(u_{5}\)
\(u_{n}\)
Explanation
This means that for arithmetic sequences: $$u_{n}=u_{1}+\left(n-1\right)d$$
Examples and Your Turns
Example
Find the \(10\)th term of the arithmetic sequence \(19, 25, 31, 37, …\)
Your Turn
Find the \(15\)th term of the arithmetic sequence \(5, -3, -11, -19, …\)
Example
The second term of an arithmetic sequence is \(15\) and the fifth term is \(21\). Find the \(12\)th term of the sequence.
Your Turn
For an arithmetic sequence we know that \(u_{3}=8\) and \(u_{8}=-17\). Find the value of \(u_{24}\).
Your Turn
Find the number of terms in the arithmetic sequence \(50, 47, 44, …, 14\).
Your Turn
An arithmetic progression has first term \(5\) and common difference \(7\). What is the term number of \(355\) in this progression?
Your Turn
Find \(k\) given that \(3k+1, k\) and \(-3\) are consecutive terms in an arithmetic sequence.
Your Turn
Three numbers are consecutive terms in an arithmetic sequence. Their sum is \(48\) and their product is \(2800\). Find the three numbers.
Notes
Consider this problem: what is the sum of the integers from \(1\) to \(100\)?
Consider the general arithmetic series with first term \(u_{1}\) and common difference \(d\).
$$\begin{matrix} S_{n} & = & u_{1} & + & u_{2} & + & u_{3} & + & u_{4} & + & … & + & u_{n} \\ S_{n} & = & u_{1} & + & u_{1}+d & + & \qquad & + & \qquad & + & … & + & \qquad\end{matrix}$$
If we write this series twice, once in order and once with the terms in reverse we get:
$$\begin{matrix} S_{n} & = & u_{1} & + & u_{1}+d & + & \qquad & + & \qquad & + & … & + & \qquad \\ S_{n} & = & \qquad & + & \qquad & + & \qquad & + & \qquad & + & … & + & u_{1}\end{matrix}$$
Adding these two equations together we get
$$2S_{n} = \underbrace{\begin{matrix}\qquad & + & \qquad & + & \qquad & + & \qquad & + & … & + & \qquad \end{matrix}}_{\qquad \text{times}}$$
This leads us to the following formulae for arithmetic series: $$S_{n}=\frac{n}{2}\left(u_{1}+u_{n}\right)=\frac{n}{2}\left(2u_{1}+\left(n-1\right)d\right)$$
Examples and Your Turns
Example
The first term of an arithmetic sequence is \(2\) and the last term is \(26\). The series has \(9\) terms. Find the sum of the series.
Your Turn
Find the sum of the series $$-6+1+8+15+…+141$$
Your Turn
Consider the arithmetic sequence which starts $$2x+3y, 3x+2y,4x+y, …$$
Determine the sum of the first \(7\) terms of this series.
Your Turn
The first term of an arithmetic sequence is \(25\) and the fourth term is \(13\). The sum of the series is \(-119\).
Find the number of terms in the series.
Your Turn
Consider the sequence of odd numbers \(1, 3, 5, 7, 9, …\)
a) State the general term \(u_{n}\) for this sequence.
b) Evaluate the following:
\(S_{1}\)
\(S_{2}\)
\(S_{3}\)
\(S_{4}\)
\(S_{5}\)
c) What do you notice about your results? Make a conjecture.
d) Prove your conjecture using the formula for the sum of an arithmetic series.
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.