In this chapter you lot will learn to create, recognise, describe, extend and make generalisations about numeric and geometric patterns. Patterns allow the states to brand predictions. You volition besides piece of work with dissimilar representations of patterns, such as menstruation diagrams and tables.

The term-term relationship in a sequence

Going from one term to the next

A list of numbers which form a design is called a sequence. Each number in a sequence is called a term of the sequence. The first number is the first term of the sequence.

Write down the next three numbers in each of the sequences below. Also explain in writing, in each case, how you figured out what the numbers should be.

  1. Sequence A: 2; 5; 8; 11; fourteen; 17; xx; 23;
  2. Sequence B: four; 5; 8; 13; twenty; 29; 40;
  3. Sequence C: one; 2; 4; 8; sixteen; 32; 64;
  4. Sequence D: 3; 5; seven; 9; 11; thirteen; xv; 17; 19;
  5. Sequence E: 4; 5; 7; 10; 14; nineteen; 25; 32; xl;
  6. Sequence F: 2; half-dozen; 18; 54; 162; 486;
  7. Sequence G: 1; five; nine; 13; 17; 21; 25; 29; 33;
  8. Sequence H: ii; 4; 8; 16; 32; 64;

Numbers that follow one another are said to be consecutive.

Adding or subtracting the aforementioned number

  1. Which sequences on the previous section are of the same kind as sequence A? Explain your answer.

    Amanda explains how she figured out how to continue sequence A:

    I looked at the first two numbers in the sequence and saw that I needed 3 to go from 2 to five. I looked further and saw that I also needed 3 to get from v to 8. I tested that and it worked for all the side by side numbers.

    This gave me a dominion I could employ to extend the equence: add 3 to each number to discover the next number in the blueprint.

    Tamara says you tin also discover the pattern by working backwards and subtracting 3 each time:

    When the differences between consecutive terms of a sequence are the same, we say the difference is constant.

    \[xiv - 3 = xi; 11 - 3 = viii; 8 - iii = 5; 5 - 3 = 2\]

  2. Provide a dominion to depict the relationship between the numbers in the sequence. Utilize this rule to calculate the missing numbers in the sequence.
    1. 1; viii; xv;______;______;______;______;______;...
    2. ten 020;______;______;______; nine 980; 9 970;______; 9 940; 9 930; ...
    3. one,five; iii,0; 4,v;______;______;______;______;______;...
    4. 2,2; 4,0; five,8;______;______;______;______;______;...
    5. \( 45; \frac{3}{4}; 46; \frac{3}{4}; 47; \frac{ane}{ii}; 48;\text{______;______;______;______;______;} \)...
    6. ______; 100,49; 100,38; 100,27; ______;______; 99,94; 99,83; 99,72;...
  3. Complete the tabular array below.

    Input number

    1

    ii

    3

    iv

    v

    12

    n

    Input number + 7

    8

    11

    15

    xxx

Multiplying or dividing with the same number

Take another expect at sequence F: 2; vi; 18; 54; 162; 486; ...

Piet explains that he figured out how to continue the sequence F:

I looked at the first two terms in the sequence and wrote \(2 \times ? = half dozen\).

When I multiplied the first number by 3, I got the second number: \(2 \times 3 = half dozen\).

I and then checked to come across if I could observe the next number if I multiplied half-dozen past 3: \(half-dozen \times 3 = 18\).

I continued checking in this way: \( 18 \times 3 = 54; 54 \times 3 = 162\) and so on.

This gave me a rule I tin can employ to extend the sequence and my dominion was: multiply each number by 3 to calculate the next number in the sequence.

Zinhle says you can also observe the design by working backwards and dividing past 3 each time:

\[ 54 \div 3 = 18; 18 \div 3 = 6;6 \div 3= 2\]

The number that we multiply with to get the next term in the sequence is called a ratio. If the number nosotros multiply with remains the same throughout the sequence, we say information technology is a constant ratio.

  1. Check whether Piet's reasoning works for sequence H: 2; 4; 8; 16; 32; 64; ...
  2. Depict, in words, the rule for finding the side by side number in the sequence. Also write downwardly the next five terms of the sequence if the design is connected.
    1. i; 10; 100; 1 000;
    2. 16; 8; iv; 2;
    3. vii; -21; 63; -189;
    4. 3; 12, 48;
    5. ii 187; -729; 243; -81;
    1. Fill in the missing output and input numbers:

      72618.png

      What is the term-to-term rule for the output numbers here, \(+ vi \text{ or } \times 6?\)

    2. Complete the table beneath:

      Input numbers

      1

      2

      3

      iv

      v

      12

      x

      Output numbers

      6

      24

      36

Neither adding nor multiplying past the aforementioned number

  1. Consider sequences A to H once more and answer the questions that follow:

    Sequence A: 2; 5; 8; 11; 14; 17; 20; 23; ...

    Sequence B: 4; v; 8; 13; 20; 29; forty;...

    Sequence C: 1; ii; four; eight; 16; 32; 64;...

    Sequence D: 3; 5; seven; 9; 11; thirteen; xv; 17; 19; ...

    Sequence E: 4; 5; seven; x; fourteen; 19; 25; 32; twoscore;...

    Sequence F: 2; vi; 18; 54; 162; 486;...

    Sequence G: 1; five; 9; 13; 17; 21; 25; 29; 33;...

    Sequence H: two; 4; 8; 16; 32; 64;...

    1. Which other sequence(s) is/are of the aforementioned kind as sequence B? Explicate.
    2. In what way are sequences B and E dissimilar from the other sequences?

At that place are sequences where there is neither a constant difference nor a constant ratio betwixt sequent terms and yet a pattern still exists, equally in the case of sequences B and Eastward.

  1. Consider the sequence: 10; 17; 26; 37; 50; ...
    1. Write down the adjacent v numbers in the sequence.
    2. Eric observed that he tin calculate the next term in the sequence equally follows: 10 + 7 = 17; 17 + 9 = 26; 26 + 11 = 37. Use Eric's method to check whether your numbers in question (a) above are right.
  2. Which of the statements below can Eric apply to depict the relationship betwixt the numbers in the sequence in question 2? Test the rule for the commencement iii terms of the sequence and then but write "yeah" or "no" next to each statement.
    1. Increase the difference between consecutive terms by ii each fourth dimension
    2. Increase the difference between consecutive terms by 1 each time
    3. Add two more than you added to get the previous term
  3. Provide a rule to describe the human relationship between the numbers in the sequences beneath. Use your rule to provide the next five numbers in the sequence.
    1. ane; iv; 9; 16; 25;
    2. 2; 13; 26; 41; 58;
    3. four; 14; 29; 49; 74;
    4. 5; half-dozen; 8; eleven; 15; 20;

The position-term relationship in a sequence

Using position to make predictions

  1. Take some other look at equences A to H. Which sequence(s) are of the aforementioned kind equally sequence A? Explain.

    Sequence A: 2; 5; eight; eleven; 14; 17; twenty; 23;...

    Sequence B: 4; 5; viii; xiii; twenty; 29; xl;...

    Sequence C: 1; 2; 4; 8; 16; 32; 64;...

    Sequence D: 3; 5; seven; 9; 11; thirteen; xv; 17; 19;...

    Sequence E: 4; five; vii; 10; 14; 19; 25; 32; 40;...

    Sequence F: 2; vi; 18; 54; 162; 486; ...

    Sequence G: ane; 5; ix; thirteen; 17; 21; 25; 29; 33;...

    Sequence H: two; iv; 8; sixteen; 32; 64;...

Sizwe has been thinking virtually Amanda and Tamara's explanations of how they worked out the rule for sequence A and has drawn up a table. He agrees with them only says that in that location is some other rule that will also work. He explains:

My table shows the terms in the sequence and the difference betwixt consecutive terms:

1st term

2nd term

tertiary term

fourth term

A:

5

8

xi

14

differences

+3

+3

+3

+3

+3

+iii

+3

+iii

+3

Sizwe reasons that the following rule will besides piece of work:

Multiply the position of the number by iii and add together 2 to the answer.

I can write this rule as a number sentence: Position of the number\(\bf{ \times 3 + 2}\)

I use my number judgement to check: \( {\bf1} \times 3 + two = 5; {\bf2} \times iii + 2 = 8; {\bf3} \times 3 + ii = 11 \)

    1. What exercise the numbers in bold in Sizwe's number sentence stand up for?
    2. What does the number iii in Sizwe's number sentence stand for?
  2. Consider the sequence 5; 8; 11; 14; ...

    Apply Sizwe's dominion to the sequence and determine:

    1. term number 7 of the sequence
    2. term number 10 of the sequence
    3. the 100th term of the sequence
  3. Consider the sequence: three; 5; 7; 9; 11; 13; 15; 17; xix;..
    1. Utilise Sizwe's explanation to find a rule for this sequence.
    2. Determine the 28th term of the sequence.

More predictions

Complete the tables below past computing the missing terms.

  1. Position in sequence

    1

    2

    3

    iv

    10

    54

    Term

    4

    7

    x

    thirteen

  2. Position in sequence

    1

    2

    3

    4

    8

    16

    Term

    4

    ix

    14

    19

  3. Position in sequence

    1

    two

    3

    4

    7

    xxx

    Term

    3

    15

    27
  4. Employ the dominion Position in the sequence \(\times\) (position in the sequence + ane) to complete the table below.

    Position in sequence

    1

    two

    iii

    four

    5

    half dozen

    Term

    2

Investigating and extending geometric patterns

Square numbers

A factory makes window frames. Type 1 has one windowpane, type 2 has four windowpanes, type three has nine windowpanes, and then on.

72249.png

  1. How many windowpanes volition at that place be in blazon five?
  2. How many windowpanes will there be in type 6?
  3. How many windowpanes will there be in type seven?
  4. How many windowpanes will there be in type 12? Explain.
  5. Complete the table. Show your calculations.

    Frame type

    1

    ii

    three

    iv

    15

    20

    Number of windowpanes

    i

    4

    9

    16

The symbol n is used below to represent the position number in the expression that gives the rule (\(due north^2\)) when generalising.

In algebra we think of a square every bit a number that is obtained by multiplying a number by itself. And then 1 is too a square because \(1 \times 1 = 1\).

72098.png

Triangular numbers

Therese uses circles to form a design of triangular shapes:

72088.png

  1. If the design is connected, how many circles must Therese take
    1. in the bottom row of picture show 5?
    2. in the second row from the lesser of motion-picture show five?
    3. in the third row from the bottom of film five?
    4. in the 2d row from the top of picture five?
    5. in the top row of picture 5?
    6. in total in flick 5? Prove your calculation.
  2. How many circles does Therese need to form triangle movie seven? Show the calculation.
  3. How many circles does Therese need to form triangle pic 8?
  4. Consummate the table below. Show all your work.

    Moving picture number

    1

    ii

    3

    four

    5

    6

    12

    fifteen

    Number of circles

    1

    three

    6

    10

More than than 2 500 years ago, Greek mathematicians already knew that the numbers 3, 6, ten, 15 and so on could form a triangular pattern. They represented these numbers with dots which they bundled in such a fashion that they formed equilateral triangles, hence the name triangular numbers. Algebraically nosotros think of them as sums of consecutive natural numbers starting with 1.

Let us revisit the action on triangular numbers that we did in the previous section.

71938.png

So far, we take determined the number of circles in the pattern by calculation consecutive natural numbers. If nosotros were asked to determine the number of circles in flick 200, for example, it would take us a very long time to do so. We need to discover a quicker method of finding any triangular number in the sequence.

Consider the system below.

71929.png

We accept added the yellow circles to the original blueish circles then rearranged the circles in such a way that they are in a rectangular form.

  1. Pic 2 is 3 circles long and 2 circles wide. Consummate the post-obit sentences:
    1. Moving-picture show 3 is ______ circles long and ______ circles wide.
    2. Film 1 is ______ circles long and ______ circle wide.
    3. Picture iv is ______ circles long and ______ circles wide.
    4. Film 5 is ______ circles long and ______ circles broad.
  2. How many circles will there exist in a picture that is:
    1. 10 circles long and 9 circles wide?
    2. seven circles long and half dozen circles wide?
    3. 6 circles long and v circles wide?
    4. 20 circles long and nineteen circles wide?

Suppose we desire to have a quicker method to determine the number of circles in picture 15. Nosotros know that picture fifteen is 16 circles long and fifteen circles broad. This gives a full of \(15 \times 16 = 240\) circles. Merely nosotros must compensate for the fact that the yellow circles were originally not at that place by halving the full number of circles. In other words, the original figure has \(240 \div 2 = 120\) circles.

  1. Use the above reasoning to calculate the number of circles in:
    1. picture 20
    2. picture 35

Describing patterns in different means

T-shaped numbers

The pattern beneath is made from squares.

71576.png

    1. How many squares will there exist in pattern 5?
    2. How many squares will there be in pattern 15?
    3. Complete the table.

      Blueprint number

      1

      2

      3

      4

      5

      6

      20

      Number of squares

      1

      4

      7

      10

Below are three dissimilar methods or plans to summate the number of squares for pattern 20. Report each one carefully.

Plan A:

To get from 1 square to 4 squares, you have to add 3 squares. To become from 4 squares to 7 squares, you have to add iii squares. To get from 7 squares to 10 squares, you accept to add iii squares. So continue to add together iii squares for each design until pattern xx.

Program B:

Multiply the pattern number by 3, and decrease 2. So pattern xx will have \(20 \times three - ii\) squares.

Plan C:

The number of squares in pattern 5 is 13. So blueprint 20 will have \(13 \times four = 52\) squares because \(xx = v \times iv\).

    1. Which method or program (A, B or C) will give the right answer? Explain why.
    2. Which of the above plans did you lot apply? Explain why?
    3. Tin this menstruum diagram be used to calculate the number of squares?

      71484.png

... and some other shapes

  1. Three figures are given beneath. Draw the side by side effigy in the tile pattern.

    71475.png

    1. If the pattern is connected, how many tiles will there be in the 17th figure? Answer this question by analysing what happens.
    2. Thato decides that information technology easier for him to encounter the pattern when the tiles are rearranged equally shown here:

      71462.png

      Utilise Thato's method to determine the number of tiles in the 23rd figure.

    3. Complete the period diagram beneath by writing the appropriate operators so that it tin be used to calculate the number of tiles in any figure of the pattern.

      71453.png

    4. How many tiles will there be in the 50th effigy if the pattern is continued?
  1. Write down the next four terms in each sequence. Besides explain, in each instance, how you figured out what the terms are.
    1. 2; 4; eight; 14; 22; 32; 44;
    2. ii; 6; 18; 54; 162;
    3. 1; 7; 13; 19; 25;
    1. Complete the table below by calculating the missing terms.

      Position in sequence

      1

      2

      3

      4

      5

      seven

      x

      Term

      3

      10

      17
    2. Write the rule to calculate the term from the position in the sequence in words.
  3. Consider the stacks below.

    71385.png

    1. How many cubes will there exist in stack 5?
    2. Complete the table.

      Stack number

      i

      2

      iii

      four

      5

      vi

      10

      Number of cubes

      one

      8

      27
    3. Write downwards the rule to calculate the number of cubes for whatsoever stack number.