Unit 1 - Number and Algebra
Required Prior Knowledge
Questions
Solve this pair of simultaneous equations.
$$\begin{matrix}xy & = & 24 \\ xy^{5} & = & 6144 \end{matrix}$$
Solutions
Get Ready
Questions
What is the same and what is different about these three sequences?
$$3, 15, 75, 375, …$$
$$u_{1}=3, u_{n+1}=8u_{n}$$
$$a_{n}=3\times 2^{n-1}$$
Solutions
Notes
A geometric sequence is a sequence where each successive term is the previous term multiplied by some fixed value.
This fixed value is known as the common ratio and is usually denoted with \(r\).
A sequence is geometric if and only if \(\frac{u_{n+1}}{u_{n}}=r\) for all \(n\in\mathbb{Z}^{+}\).
That is, if $$\frac{u_{2}}{u_{1}}=\frac{u_{3}}{u_{2}}=\frac{u_{4}}{u_{3}}=r$$
Note that if \(r<0\) then the sequence will oscillate about \(0\).
For example, find the common ratio for each of these geometric sequences:
a) \(10, 25, 62.5, …\)
b) \(\frac{1}{2},-\frac{1}{6},\frac{1}{18},…\)
c) \(a, 2a^{3}, 4a^{5},…\)
-
a) \(r=\frac{25}{10}=\frac{62.5}{25}=2.5\)
b) \(r=\frac{-\frac{1}{6}}{\frac{1}{2}}=\frac{\frac{1}{18}}{-\frac{1}{6}}=-\frac{1}{3}\)
c) \(r=\frac{2a^{3}}{a}=\frac{4a^{5}}{2a^{3}}=2a^{2}\)
Consider the geometric sequence with \(u_{1}=3\) and \(r=2\).
Work out:
\(u_{2}\)
\(u_{3}\)
\(u_{4}\)
\(u_{5}\)
\(u_{n}\)
-
\(u_{2}=3\times 2=6\)
\(u_{3}=6\times 2=12\)
\(u_{4}=12\times 2=24\)
\(u_{5}=24\times 2=48\)
\(u_{n}=u_{n-1}\times 2=3\times 2^{n-1}\)
Now consider the geometric sequence with \(u_{1}=u_{1}\) and \(r=r\).
Work out:
\(u_{2}\)
\(u_{3}\)
\(u_{4}\)
\(u_{5}\)
\(u_{n}\)
Explanation
This means that for geometric sequences: $$u_{n}=u_{1}\times r^{n-1}$$
Examples and Your Turns
Example
Find the \(12\)th term of the geometric sequence \(16, 8, 4, 2, …\)
Your Turn
\(k-1, 2k\) and \(21-k\) are three consecutive terms in a geometric sequence. Find \(k\).
Your Turn
A geometric sequence has \(u_{2}=-6\) and \(u_{5}=162\). Find the general form of the sequence.
Your Turn
The seventh term of a geometric sequence is \(13\). The ninth term is \(52\). Find the possible values of the fourth term.
Your Turn
A geometric sequence has first term \(10\) and common ratio \(\frac{1}{3}\). What is the first term that is less than \(10^{-6}\)?
Your Turn
A geometric sequence has first term \(2\) and common ratio \(-3\). What term has the value \(-4374\)?
Notes
One grain of rice is placed on the first square of a chessboard.
Two grains of rice are placed on the second square of a chessboard.
Four grains of rice are placed on the third square of a chessboard.
And so on, doubling the number of grains of rice for each subsequent square.
How many grains of rice are there on the chessboard in total?
Consider the general geometric series with first term \(u_{1}\) and common ratio \(r\).
$$\begin{matrix} S_{n} & = & u_{1} & + & u_{2} & + & u_{3} & + & u_{4} & + & … & + & u_{n} \\ S_{n} & = & u_{1} & + & u_{1}r & + & \qquad & + & \qquad & + & … & + & \qquad\end{matrix}$$
If we multiply this whole equation by \(r\) we get:
$$\begin{matrix} S_{n} & = & u_{1} & + & u_{1}r & + & \qquad & + & \qquad & + & … & + & \qquad \\ rS_{n} & = & u_{1}r & + & \qquad & + & \qquad & + & \qquad & + & … & + & \qquad\end{matrix}$$
Shifting the second equation along we get:
$$\begin{matrix} S_{n} & = & u_{1} & + & u_{1}r & + & \qquad & + & \qquad & + & … & + & \qquad \\ rS_{n} & = & \qquad & \qquad & u_{1}r & + & \qquad & + & \qquad & + & \qquad & + & … & + & \qquad\end{matrix}$$
This leads us to the following formulae for geometric series: $$S_{n}=\frac{u_{1}\left(r^{n}-1\right)}{r-1}=\frac{u_{1}\left(1-r^{n}\right)}{1-r}$$
Examples and Your Turns
Example
For the series \(2+6+18+54+…\) find the value of \(S_{12}\).
Your Turn
Find the exact value of the sum of the first six terms of the geometric series with first term \(8\) and common ratio \(\frac{1}{2}\)
Your Turn
Find the sum of the series $$81-27+9-3+…+\frac{1}{729}$$
Your Turn
How many terms are needed for the sum of the geometric series \(3+6+12+24+…\) to exceed \(10,000\).
Your Turn
Calculate the geometric series given by $$\sum^{6}_{r=1}\left(3\times 2^{r}\right) $$
Your Turn
Find two possible geometric sequences where the sum of the first two terms is \(9\) and the sum of the first four terms is \(45\) and write down the general term of each sequence.
Your Turn
The sum of the first \(n\) terms of a geometric series is given by \(S_{n}=5^{n}-1\). Find the first term and the common ratio of this series.
Notes
Let’s consider different values of \(r\) in the expression \(r^{n}\) as \(n\) tends to infinity.
Fill in the table below with the value of each sequence for the different values of \(n\).
When \(r>1\), the series
When \(0<r<1\), the series
When \(-1<r<0\), the series
When \(r<-1\), the series
When \(r=0\), the series
When \(r=\pm 1\), the series
-
When \(r>1\), the series diverges.
When \(0<r<1\), the series converges to \(0\).
When \(-1<r<0\), the series converges to \(0\).
When \(r<-1\), the series diverges and oscillates.
When \(r=0\), the series is always \(0\).
When \(r=\pm 1\), the series is constant or oscillates between positive and negative.
-
When \(r>1\), as \(n\) gets bigger, \(r^{n}\rightarrow \infty\).
When \(0<r<1\), as \(n\) gets bigger, \(r^{n}\rightarrow 0\).
When \(-1<r<0\), as \(n\) gets bigger, \(r^{n}\rightarrow 0\).
When \(r<-1\), as \(n\) gets bigger, \(\left|r^{n}\right|\rightarrow \infty\).
Now consider what happens to the formula below in each scenario as \(n\rightarrow \infty\): $$S_{n}=\frac{u_{1}\left(1-r^{n}\right)}{1-r}$$
So, for a geometric series where \(0<\left| r\right|<1\), we can calculate the sum to infinity using $$S_{\infty}=\frac{u_{1}}{1-r}$$
Examples and Your Turns
Example
Find the value of the sum to infinity of this geometric series $$0.7+0.07+0.007+0.0007+…$$
Your Turn
Find the sum to infinity of the series $$1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+…$$
Your Turn
The sum to infinity of a geometric series is \(5\). The second term is \(-\frac{6}{5}\). Find the common ratio.
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.
Taking it Deeper
Conceptual Questions to Consider
How can you determine if a given sequence is geometric?
Compare and contrast arithmetic and geometric sequences. How are they similar? How are they different?
Explain why a geometric series with \(\left|r\right|\ge 1\) does not converge.
Common Mistakes / Misconceptions
Using \(d\) instead of \(r\). That is, confusing arithmetic and geometric sequences.
Forgetting that \(\left|r\right|<1\) is an important condition to be able to calculate \(S_{\infty}\).
Connecting This to Other Skills
This skill builds upon the ideas introduced in 1.3 Sequences, Series and Sigma notation.
The next skill where we look at compound interest (1.6) is an application of geometric sequences and series.
In skill 1.18 we will look at using the binomial theorem for rational powers, and this depends upon the idea of converging geometric series.
When we start to look at limits in 4.2 and unit 8, these build upon the ideas of convergence first seen here.
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?