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You are watching: The next number in this series 7, 9, 11, 13, 15, is:

In this thing you will learn to create, recognise, describe, extend and also make generalisations around numeric and also geometric patterns. Patterns permit us to make predictions. Friend will additionally work with different representations of patterns, together as flow diagrams and tables.

## The term-term partnership in a sequence

### Going indigenous one term come the next

A list of number which kind a sample is called a sequence. Every number in a sequence is referred to as a term the the sequence. The very first number is the very first term that the sequence.

Write down the following three number in every of the assignment below. Additionally explain in writing, in each case, how you established what the numbers need to be.

succession A: 2; 5; 8; 11; 14; 17; 20; 23; sequence B: 4; 5; 8; 13; 20; 29; 40; sequence C: 1; 2; 4; 8; 16; 32; 64; sequence D: 3; 5; 7; 9; 11; 13; 15; 17; 19; succession E: 4; 5; 7; 10; 14; 19; 25; 32; 40; succession F: 2; 6; 18; 54; 162; 486; sequence G: 1; 5; 9; 13; 17; 21; 25; 29; 33; succession H: 2; 4; 8; 16; 32; 64;

### Adding or individually the exact same number

i m sorry sequences on the previous section are that the exact same kind as sequence A? explain your answer.

Amanda describes how she figured out how to proceed sequence A:

I looked in ~ the first two numbers in the sequence and also saw the I essential 3 to walk from 2 come 5. I looked further and saw that I also needed 3 to walk from 5 come 8. Ns tested that and it operated for every the next numbers.

This gave me a preeminence I could use to extend the equence: add 3 to each number to find the next number in the pattern.

Tamara claims you can likewise find the sample by working backwards and also subtracting 3 each time:

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

\<14 - 3 = 11; 11 - 3 = 8; 8 - 3 = 5; 5 - 3 = 2\>

provide a ascendancy to describe the relationship in between the number in the sequence. Usage this dominance to calculation the missing numbers in the sequence. 1; 8; 15;______;______;______;______;______;... 10 020;______;______;______; 9 980; 9 970;______; 9 940; 9 930; ... 1,5; 3,0; 4,5;______;______;______;______;______;... 2,2; 4,0; 5,8;______;______;______;______;______;... $$45; \frac34; 46; \frac34; 47; \frac12; 48;\text______;______;______;______;______;$$... ______; 100,49; 100,38; 100,27; ______;______; 99,94; 99,83; 99,72;... finish the table below.
 Input number 1 2 3 4 5 12 n Input number + 7 8 11 15 30

### Multiplying or dividing with the very same number

Take one more look at sequence F: 2; 6; 18; 54; 162; 486; ...

Piet explains that he established how to continue the succession F:

I looked at the first two state in the sequence and wrote $$2 \times ? = 6$$.

When i multiplied the first number by 3, I gained the second number: $$2 \times 3 = 6$$.

I then confirm to view if I might find the following number if ns multiplied 6 through 3: $$6 \times 3 = 18$$.

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

This provided me a dominance I deserve to use to expand the sequence and my ascendancy was: multiply every number through 3 to calculation the next number in the sequence.

Zinhle claims you can additionally find the pattern by functioning backwards and dividing by 3 every time:

\< 54 \div 3 = 18; 18 \div 3 = 6;6 \div 3= 2\>

The number that we multiply v to get the following term in the succession is called a ratio. If the number us multiply with continues to be the very same throughout the sequence, we say that is a constant ratio.

inspect whether Piet"s thinking works for sequence H: 2; 4; 8; 16; 32; 64; ... Describe, in words, the ascendancy for detect the next number in the sequence. Likewise write down the next five terms the the sequence if the pattern is continued. 1; 10; 100; 1 000; 16; 8; 4; 2; 7; -21; 63; -189; 3; 12, 48; 2 187; -729; 243; -81; fill in the missing output and input numbers:

What is the term-to-term dominance for the calculation numbers here, $$+ 6 \text or \times 6?$$

complete the table below:
 Input numbers 1 2 3 4 5 12 x Output numbers 6 24 36

### Neither adding nor multiply by the very same number

consider sequences A to H again and also answer the inquiries that follow:

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

Sequence B: 4; 5; 8; 13; 20; 29; 40;...

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

Sequence D: 3; 5; 7; 9; 11; 13; 15; 17; 19; ...

Sequence E: 4; 5; 7; 10; 14; 19; 25; 32; 40;...

Sequence F: 2; 6; 18; 54; 162; 486;...

Sequence G: 1; 5; 9; 13; 17; 21; 25; 29; 33;...

Sequence H: 2; 4; 8; 16; 32; 64;...

Which various other sequence(s) is/are of the exact same kind as sequence B? Explain. In what method are sequences B and also E various from the various other sequences?

There space sequences whereby there is no a constant difference nor a continuous ratio in between consecutive terms and also yet a sample still exists, as in the instance of assignment B and also E.

think about the sequence: 10; 17; 26; 37; 50; ... write down the next five numbers in the sequence. Eric observed that he have the right to calculate the following term in the sequence as follows: 10 + 7 = 17; 17 + 9 = 26; 26 + 11 = 37. Use Eric"s technique to inspect whether her numbers in concern (a) over are correct. which of the statements listed below can Eric usage to explain the relationship in between the numbers in the succession in question 2? check the dominance for the very first three regards to the sequence and also then merely write "yes" or "no" beside each statement. increase the difference between consecutive state by 2 each time boost the difference in between consecutive terms by 1 each time add two more than you added to get the previous ax provide a dominance to describe the relationship in between the numbers in the assignment below. Use your ascendancy to administer the next 5 numbers in the sequence. 1; 4; 9; 16; 25; 2; 13; 26; 41; 58; 4; 14; 29; 49; 74; 5; 6; 8; 11; 15; 20;

## The position-term partnership in a sequence

### Using place to make predictions

Take an additional look in ~ equences A come H. Which sequence(s) space of the exact same kind together sequence A? Explain.

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

Sequence B: 4; 5; 8; 13; 20; 29; 40;...

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

Sequence D: 3; 5; 7; 9; 11; 13; 15; 17; 19;...

Sequence E: 4; 5; 7; 10; 14; 19; 25; 32; 40;...

Sequence F: 2; 6; 18; 54; 162; 486; ...

Sequence G: 1; 5; 9; 13; 17; 21; 25; 29; 33;...

Sequence H: 2; 4; 8; 16; 32; 64;...

Sizwe has actually been thinking around Amanda and also Tamara"s explanations of exactly how they resolved the dominance for sequence A and has attracted up a table. That agrees through them however says that there is one more rule the will likewise work. He explains:

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

A:

differences

 1st term 2nd term 3rd term 4th term 5 8 11 14 +3 +3 +3 +3 +3 +3 +3 +3 +3

Sizwe reasons that the following dominion will additionally work:

Multiply the position of the number through 3 and add 2 to the answer.

I deserve to write this dominance as a number sentence: Position of the number$$\bf \times 3 + 2$$

I usage my number sentence come check: $$\bf1 \times 3 + 2 = 5; \bf2 \times 3 + 2 = 8; \bf3 \times 3 + 2 = 11$$

What execute the numbers in interlocutor in Sizwe"s number sentence stand for? What does the number 3 in Sizwe"s number sentence was standing for? think about the succession 5; 8; 11; 14; ...

Apply Sizwe"s rule to the sequence and determine:

hatchet number 7 the the sequence ax number 10 that the sequence the 100th term of the sequence think about the sequence: 3; 5; 7; 9; 11; 13; 15; 17; 19;.. use Sizwe"s explanation to find a preeminence for this sequence. recognize the 28th term of the sequence.

### More predictions

Complete the tables listed below by calculating the absent terms.

Position in sequence

 1 2 3 4 10 54 Term 4 7 10 13
 Position in sequence 1 2 3 4 8 16 Term 4 9 14 19
 Position in sequence 1 2 3 4 7 30 Term 3 15 27
usage the dominance Position in the sequence $$\times$$ (position in the sequence + 1) to finish the table below.
 Position in sequence 1 2 3 4 5 6 Term 2

## Investigating and also extending geometric patterns

### Square numbers

A manufacturing facility makes window frames. Form 1 has one windowpane, type 2 has 4 windowpanes, kind 3 has actually nine windowpanes, and also so on.

### Triangular numbers

Therese offers circles to type a pattern of triangle shapes:

If the sample is continued, how many circles should Therese have in the bottom heat of picture 5? in the second row from the bottom of picture 5? in the third row indigenous the bottom of photo 5? in the second row from the peak of snapshot 5? in the peak row of photo 5? in total in photo 5? display your calculation. How plenty of circles walk Therese need to type triangle picture 7? display the calculation. How numerous circles go Therese need to kind triangle snapshot 8? complete the table below. Present all your work.
 Picture number 1 2 3 4 5 6 12 15 Number that circles 1 3 6 10

More 보다 2 500 years ago, Greek mathematicians already knew that the number 3, 6, 10, 15 and also so ~ above could kind a triangular pattern. They stood for these numbers with dots i m sorry they arranged in such a means that they developed equilateral triangles, for this reason the name triangular numbers. Algebraically we think of them as sums the consecutive organic numbers starting with 1.

Let united state revisit the activity on triangular number that we did in the ahead section.

So far, us have established the variety of circles in the sample by including consecutive herbal numbers. If we were request to determine the variety of circles in snapshot 200, for example, it would take us a very long time to carry out so. We need to discover a quicker technique of finding any type of triangular number in the sequence.

Consider the setup below.

We have added the yellow one to the initial blue circles and also then rearranged the circles in such a means that they room in a rectangle-shaped form.

picture 2 is 3 one long and 2 circles wide. Complete the adhering to sentences: snapshot 3 is ______ one long and ______ circles wide. snapshot 1 is ______ one long and also ______ circle wide. snapshot 4 is ______ one long and ______ circles wide. snapshot 5 is ______ one long and also ______ circles wide. How plenty of circles will certainly there it is in in a snapshot that is: 10 one long and also 9 one wide? 7 one long and 6 one wide? 6 circles long and also 5 one wide? 20 one long and 19 one wide?

Suppose we desire to have actually a quicker method to recognize the number of circles in photo 15. We know that photo 15 is 16 circles long and also 15 circles wide. This offers a full of $$15 \times 16 = 240$$ circles. Yet we must compensate because that the fact that the yellow circles were originally not over there by halving the total number of circles. In other words, the original figure has $$240 \div 2 = 120$$ circles.

usage the above reasoning to calculation the number of circles in: snapshot 20 picture 35

## Describing trends in various ways

### T-shaped numbers

The pattern listed below is make from squares.

How numerous squares will there be in pattern 5? How many squares will there be in pattern 15? complete the table.
 Pattern number 1 2 3 4 5 6 20 Number that squares 1 4 7 10

Below are three different methods or to plan to calculation the variety of squares because that pattern 20. Research each one carefully.

Plan A:

To acquire from 1 square come 4 squares, you have actually to include 3 squares. To get from 4 squares come 7 squares, you have to add 3 squares. To obtain from 7 squares to 10 squares, you have actually to add 3 squares. So continue to add 3 squares because that each pattern till pattern 20.

Plan B:

Multiply the sample number through 3, and subtract 2. So pattern 20 will have actually $$20 \times 3 - 2$$ squares.

Plan C:

The variety of squares in sample 5 is 13. So pattern 20 will have $$13 \times 4 = 52$$ squares due to the fact that $$20 = 5 \times 4$$.

See more: What Is The Square Root Of 183 In Simplest Radical Form? How To Simplify Square Root Of 183

Which technique or setup (A, B or C) will provide the right answer? explain why. which of the over plans did friend use? define why? can this flow diagram be used to calculation the variety of squares?

### ... And some other shapes

Three figures are provided below. Attract the next number in the tile pattern.

If the pattern is continued, how plenty of tiles will certainly there it is in in the 17th figure? prize this inquiry by analysing what happens. Thato decides the it simpler for that to watch the pattern as soon as the tiles room rearranged as presented here:

Use Thato"s technique to determine the number of tiles in the 23rd figure.

finish the flow diagram listed below by writing the ideal operators so that it have the right to be supplied to calculate the number of tiles in any figure of the pattern.

How many tiles will there be in the 50th figure if the pattern is continued?
compose down the next four terms in each sequence. Also explain, in every case, exactly how you identified what the terms are. 2; 4; 8; 14; 22; 32; 44; 2; 6; 18; 54; 162; 1; 7; 13; 19; 25; finish the table listed below by calculating the missing terms.
 Position in sequence 1 2 3 4 5 7 10 Term 3 10 17
compose the rule to calculate the term native the position in the succession in words. think about the stacks below.

How many cubes will there it is in in stack 5? finish the table.

Stack number

 1 2 3 4 5 6 10 Number the cubes 1 8 27
create down the preeminence to calculate the number of cubes for any stack number.