# 2.1 Notation

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## Notation

The ${n}^{\mathrm{th}}$ -term of a sequence is written as ${a}_{n}$ . So for example, the ${1}^{\mathrm{st}}$ -term of a sequence is ${a}_{1}$ , the ${10}^{\mathrm{th}}$ -term is ${a}_{10}$ .

A sequence does not have to follow a pattern but when it does, we can often write down a formula to calculate the ${n}^{\mathrm{th}}$ -term, ${a}_{n}$ . In the sequence

$1;4;9;16;25;...$

where the sequence consists of the squares of integers, the formula for the ${n}^{\mathrm{th}}$ -term is

$\begin{array}{c}\hfill {a}_{n}={n}^{2}\end{array}$

You can check this by looking at:

$\begin{array}{ccc}\hfill {a}_{1}& =& {1}^{2}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}1\hfill \\ \hfill {a}_{2}& =& {2}^{2}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}4\hfill \\ \hfill {a}_{3}& =& {3}^{2}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}9\hfill \\ \hfill {a}_{4}& =& {4}^{2}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}16\hfill \\ \hfill {a}_{5}& =& {5}^{2}\phantom{\rule{0.277778em}{0ex}}=\phantom{\rule{0.277778em}{0ex}}25\hfill \\ \hfill ...\end{array}$

Therefore, using [link] , we can generate a pattern, namely squares of integers.

We can also define the common difference for a pattern.

Common difference
The common difference is the difference between successive terms and is denoted by d.
For example, consider the sequence $10;7;4;1;\mathrm{...}$ . To find the common difference, we simply subtract each successive term:
$\begin{array}{ccc}7-10& =& -3\\ 4-7& =& -3\\ 1-4& =& -3\end{array}$

As before, you and 3 friends are studying for Maths, and you are seated at a square table. A few minutes later, 2 other friends join you and add another table to the existing one. Now 6 of you can sit together. A short time later 2 more of your friends join your table, and you add a third table to the existing tables. Now 8 of you can sit comfortably as shown: For each table added, two more people can be seated.

Find the expression for the number of people seated at $n$ tables. Then, use the general formula to determine how many people can sit around 12 tables and how many tables are needed for 20 people.

1.  Number of Tables , $n$ Number of people seated Formula 1 $4=4$ $=4+2·\left(0\right)$ 2 $4+2=6$ $=4+2·\left(1\right)$ 3 $4+2+2=8$ $=4+2·\left(2\right)$ 4 $4+2+2+2=10$ $=4+2·\left(3\right)$ $⋮$ $⋮$ $⋮$ $n$ $4+2+2+2+...+2$ $=4+2·\left(n-1\right)$
2. The number of people seated at $n$ tables is:

${a}_{n}={a}_{1}+d·\left(n-1\right)$

Notice how we have used d to represent the common difference. We could also have written a 2 in place of the d. We also used ${a}_{1}$ to represent the first term, rather than using the actual value (4).

3. Considering the example from the previous section, how many people can sit around say 12 tables? We are looking for ${a}_{12}$ , that is, where $n=12$ :

$\begin{array}{ccc}\hfill {a}_{n}& =& {a}_{1}+d·\left(n-1\right)\hfill \\ \hfill {a}_{12}& =& 4+2·\left(12-1\right)\hfill \\ & =& 4+2\left(11\right)\hfill \\ & =& 4+22\hfill \\ & =& 26\hfill \end{array}$
4. $\begin{array}{ccc}\hfill {a}_{n}& =& {a}_{1}+d·\left(n-1\right)\hfill \\ \hfill 20& =& 4+2·\left(n-1\right)\hfill \\ \hfill 20-4& =& 2·\left(n-1\right)\hfill \\ \hfill 16÷2& =& n-1\hfill \\ \hfill 8+1& =& n\hfill \\ \hfill n& =& 9\hfill \end{array}$
5. 26 people can be seated at 12 tables and 9 tables are needed to seat 20 people.

It is also important to note the difference between $n$ and ${a}_{n}$ . $n$ can be compared to a place holder, while ${a}_{n}$ is the value at the place “held” by $n$ . Like our “Study Table” example above, the first table (Table 1) holds 4 people. Thus, at place $n=1$ , the value of ${a}_{1}=4$ and so on:

 $n$ 1 2 3 4 ... ${a}_{n}$ 4 6 8 10 ...

## Investigation : general formula

1. Find the general formula for the following sequences and then find ${a}_{10}$ , ${a}_{50}$ and ${a}_{100}$ :
1. $2;5;8;11;14;...$
2. $0;4;8;12;16;...$
3. $2;-1;-4;-7;-10;...$
2. The general term has been given for each sequence below. Work out the missing terms.
1. $0;3;...;15;24$        ${n}^{2}-1$
2. $3;2;1;0;...;-2$        $-n+4$
3. $-11;...;-7;...;-3$        $-13+2n$

## Patterns and conjecture

In mathematics, a conjecture is a mathematical statement which appears to be true, but has not been formally proven to be true. A conjecture can be seen as an educated guess or an idea about a pattern. A conjecture can be thought of as the mathematicians way of saying I believe that this is true, but I have no proof.

For example: Make a conjecture about the next number based on the pattern $2;6;11;17;...$

The numbers increase by 4, 5, and 6.

Conjecture: The next number will increase by 7. So, it will be $17+7$ or 24.

Consider the following pattern:

$\begin{array}{ccc}\hfill {1}^{2}+1& =& {2}^{2}-2\hfill \\ \hfill {2}^{2}+2& =& {3}^{2}-3\hfill \\ \hfill {3}^{2}+3& =& {4}^{2}-4\hfill \\ \hfill {4}^{2}+4& =& {5}^{2}-5\hfill \end{array}$
1. Add another two rows to the end of the pattern.
2. Make a conjecture about this pattern. Write your conjecture in words.
3. Generalise your conjecture for this pattern (in other words, write your conjecture algebraically).
4. Prove that your conjecture is true.
1. $\begin{array}{c}\hfill {5}^{2}+5={6}^{2}-6\\ \hfill {6}^{2}+6={7}^{2}-7\end{array}$
2. Squaring a number and adding the same number gives the same result as squaring the next number and subtracting that number.

3. We have chosen to use $x$ here. You could choose any letter to generalise the pattern.

${x}^{2}+x={\left(x+1\right)}^{2}-\left(x+1\right)$
4. $\mathrm{Left side}:\phantom{\rule{3.33333pt}{0ex}}{x}^{2}+x$
$\mathrm{Right side}:\phantom{\rule{3.33333pt}{0ex}}{\left(x+1\right)}^{2}-\left(x+1\right)$
$\begin{array}{ccc}\mathrm{Right side}\hfill & =& {x}^{2}+2x+1-x-1\hfill \\ & =& {x}^{2}+x\hfill \\ & =& \mathrm{Left side}\hfill \\ \therefore \phantom{\rule{3.33333pt}{0ex}}{x}^{2}+x\hfill & =& {\left(x+1\right)}^{2}-\left(x+1\right)\hfill \end{array}$

## Summary

• There are several special sequences of numbers:
• Triangular numbers $1;3;6;10;15;21;28;36;45;...$
• Square numbers $1;4;9;16;25;36;49;64;81;...$
• Cube numbers $1;8;27;64;125;216;343;512;729;...$
• Fibonacci numbers $0;1;1;2;3;5;8;13;21;34;...$
• We represent the ${n}^{\mathrm{th}}$ -term with the notation ${a}_{n}$
• We can define the common difference of a sequence as the difference between successive terms.
• We can work out a general formula for each number pattern and use that to predict what any number in the pattern will be.

## Exercises

1. Find the ${n}^{\mathrm{th}}$ term for: $3,7,11,15,...$
2. Find the general term of the following sequences:
1. $-2;1;4;7;...$
2. $11;15;19;23;...$
3. sequence with ${a}_{3}=7$ and ${a}_{8}=15$
4. sequence with ${a}_{4}=-8$ and ${a}_{10}=10$
3. The seating in a section of a sports stadium can be arranged so the first row has 15 seats, the second row has 19 seats, the third row has 23 seats and so on. Calculate how many seats are in the row 25.
4. A single square is made from 4 matchsticks. Two squares in a row need 7 matchsticks and 3 squares in a row need 10 matchsticks. Determine:
1. the first term
2. the common difference
3. the formula for the general term
4. how many matchsticks are in a row of 25 squares
5. You would like to start saving some money, but because you have never tried to save money before, you have decided to start slowly. At the end of the first week you deposit R5 into your bank account. Then at the end of the second week you deposit R10 into your bank account. At the end of the third week you deposit R15. After how many weeks do you deposit R50 into your bank account?
6. A horizontal line intersects a piece of string at four points and divides it into five parts, as shown below. If the piece of string is intersected in this way by 19 parallel lines, each of which intersects it at four points, find the numberof parts into which the string will be divided.

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