Unit 1 - Number and Algebra
Required Prior Knowledge
Questions
State the term to term rule for the following sequences
a) \(3, 11, 19, 27, 35, … \)
b) \(3, 5, 9, 15, 23, …\)
c) \(48, 24, 12, 6, 3, …\)
Substitute \(n=3\) into each of the following expressions
d) \(n^{2}-2^{n}\)
e) \(\frac{2n}{21-n^{2}}\)
Solutions
Get Ready
Questions
Find the next three terms of the following sequences
a) \(1, 7, 4, 10, 7, 13, … \)
b) \(5, 7, 12, 19, 31, …\)
c) \(3, 4, 6, 10, …\)
d) O, T, T, F, F, …
e) \(3, 3, 5, 4, 4, 3, …\)
f) \(1, 10, 11, 100, …\)
g) \(1, 11, 21, 1211, 111221, …\)
Solutions
Notes
A sequence is an ordered set of terms.
We can describe a sequence:
By listing the terms e.g. 3, 4, 5, 6, 7, …
Using words e.g. “start at 3 and increase by 1 each term”
Using a formula e.g. \(u_{n}=n+2\)
With a graph
We use the notation \(u_n\) to represent the \(n\)th term of a sequence, where \(n\in\mathbb{Z}\).
Using this notation, \(u_{1}\) is the first term, \(u_{2}\) is the second term and so on.
Sequences can be both finite, in which case they have a fixed number of terms, or infinite, where they go on indefinitely.
Examples and Your Turns
Example
Given the sequence defined by \(b_{n}=2-\frac{1}{n^{2}}\), determine the values of \(b_{1}, b_{2}, b_{3}\) and \(b_{4}\).
Your Turn
Given the sequence defined by \(a_{n}=15-(-2)^{n}\), determine the values of \(a_{1}, a_{2}, a_{3}\) and \(a_{4}\).
Your Turn
Determine the next term and the general term for each of these sequences
a) \(43.5, 47, 50.5, 54, …\)
b) \(\frac{1}{2}, \frac{1}{6}, \frac{1}{12}, \frac{1}{20}, …\)
c) \(1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, …\)
Notes
There are several famous types of sequence that, whilst not explicitly on the syllabus, are worth being aware of.
Fibonacci Sequence
The Fibonacci Sequence is \(1, 1, 2, 3, 5, 8, 13, 21, …\) and each term is the sum of the previous two terms.
You can get different sequences in this way by starting with a different first two terms.
They can be defined recursively by \(F_{n}=F_{n-1}+F_{n-2}\), with \(F_{1}\) and \(F_{2}\) being given.
Triangle Numbers
The Triangle Numbers are \(1, 3, 6, 10, 15, …\). We start with \(1\) then add on \(2\), then add on \(3\), then \(4\) and so on.
The \(n\)th triangle number is the sum of the natural numbers up to that point.
We can define the triangle numbers recursively by \(t_{n}=t_{n-1}+n\) with \(t_{1}=1\).
Alternatively we can define it explicitly by \(t_{n}=\frac{n\left(n+1\right)}{2}\).
Arithmetic (Linear) Sequences
An arithmetic sequence is a sequence where each term is found by adding the same value to the previous term.
We can define them recursively by \(a_{n}=a_{n-1}+c\) where \(c\) is a constant.
We look at these in more details in Skill 1-4.
Geometric Sequences
A geometric sequence is a sequence where each term is found by multiplying the previous term by the same value each time.
We can define them recursively by \(g_{n}=g_{n-1}\times k\) where \(k\) is a constant.
We look at these in more details in Skill 1-5.
Quadratic Sequences
A quadratic sequence is a sequence where the differences between terms increases by a constant.
We say the second difference is constant.
For example, in the sequence \(2, 5, 10, 17, 26, …\), the differences are \(3, 5, 7, 9, …\) but these all differ by \(2\).
Notes
A series is the sum of the terms of a sequence.
That is $$S_{n}=u_{1}+u_{2}+u_{3}+…+u_{n}$$
Note that, from this definition it is always true that
\(S_{1}=u_{1}\)
\(S_{2}=u_{1}+u_{2}\)
\(S_{n}=S_{n-1}+u_{n}\)
A series can be finite, in which case it has a particular value.
A series can also be infinite, in which case it might converge to a particular value or diverge to infinity.
When a series converges, it gets closer and closer to a particular value as more terms are added on.
For example consider this series: $$ S_{\infty}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+…$$
What value does this sequence converge to?
When a series diverges the sum goes off to infinity as you add more terms on.
This is obviously the case for the following series: $$ S_{\infty}=1+2+4+8+16+…$$
But now consider the following series, which is not that different from the one above that converged: $$ S_{\infty}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}…$$
How can we determine if this converges or diverges?
It is not part of the course to determine if a given series converges or diverges, but the idea of Limits later in the course uses these ideas.
Notes
It can be cumbersome to write out a series in long hand, so we can use Sigma Notation as a shorthand for this. $$\sum_{k=1}^{n} u_{k}=u_{1}+u_{2}+u_{3}+…+u_{n}=S_{n}$$
\(k\) is known as a dummy variable as it is not in the expanded expression.
We read this statement as “The sum of \(u_{k}\) for values of \(k\) from \(1\) to \(n\)”.
Examples and Your Turns
Example
Evaluate $$\sum_{r=1}^{5} r\left(2r+1\right)$$
Your Turn
Evaluate each of these sums
a) $$\sum_{i=1}^{5} i^{4}$$
b) $$\sum_{k=2}^{7} k+1$$
c) $$\sum_{r=0}^{3} 3^{r}$$
d) $$\sum_{k=1}^{5} \frac{1}{2^{k}}$$
Your Turn
Write an expression in sigma notation for the sum \(3+6+9+12+…+60\).
Your Turn
Write an expression in sigma notation for the sum $$\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+…+\frac{99}{100}$$
Notes
There are some universal properties of series written in sigma notation.
If \(k\) is a constant and \(a_{n}\) and \(b_{n}\) are two sequences, then
$$\sum_{i=1}^{n}k=nk$$
$$\sum_{i=1}^{n}ka_{i}=k\sum_{i=1}^{n}a_{i}$$
$$\sum_{i=1}^{n}\left(a_{i}+b_{i}\right)=\sum_{i=1}^{n}a_{i}+\sum_{i=1}^{n}b_{i}$$
$$\sum_{i=1}^{n}a_{i}=\sum_{i=0}^{n-1}a_{i+1}$$
Prove each of these is true by expanding the sigma notation.
There are also some useful known results using sigma notation:
$$\sum_{i=1}^{n}i=\frac{n\left(n+1\right)}{2}$$
$$\sum_{i=1}^{n}i^{2}=\frac{n\left(n+1\right)\left(2n+1\right)}{6}$$
$$\sum_{i=1}^{n}i^{3}=\frac{n^{2}\left(n+1\right)^{2}}{4}$$
Examples and Your Turns
Example
Evaluate the sum $$\sum_{k=1}^{8}2k+1$$ using the properties of sigma notation.
Your Turn
Evaluate the sum $$\sum_{k=1}^{20}k^{2}-3k+2$$ using the properties of sigma notation.
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.