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You are watching: Read the numbers and decide what the next number should be. 11 22 13 26 15 30 17

In this thing you will learn to create, recognise, describe, extend and make generalisations around numeric and geometric patterns. Patterns enable us to make predictions. Girlfriend will likewise work with different representations the patterns, such as flow diagrams and tables.

## The term-term relationship in a sequence

### Going native one term come the next

A list of number which type a pattern is referred to as a sequence. Each number in a succession 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 numbers in every of the assignment below. Also explain in writing, in each case, how you identified what the numbers have to be.

sequence A: 2; 5; 8; 11; 14; 17; 20; 23; succession B: 4; 5; 8; 13; 20; 29; 40; succession C: 1; 2; 4; 8; 16; 32; 64; succession D: 3; 5; 7; 9; 11; 13; 15; 17; 19; sequence 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 subtracting the exact same number

i m sorry sequences top top the previous ar are of the very same kind as sequence A? explain your answer.Amanda describes how she determined how to continue sequence A:

I looked in ~ the very first two numbers in the sequence and also saw the I needed 3 to go from 2 to 5. I looked further and saw that I also needed 3 to walk from 5 come 8. I tested that and also it functioned for all the next numbers.

This offered me **a preeminence I can use to extend the equence: add 3 to every number to uncover the following number in the pattern**.

Tamara claims you can additionally find the sample by working backwards and subtracting 3 every time:

When the **differences** between consecutive terms of a sequence room the same, we say the distinction is **constant**.

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

provide a ascendancy to explain the relationship in between the number in the sequence. Usage this dominance to calculate the absent 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; ext______;______;______;______;______; )... ______; 100,49; 100,38; 100,27; ______;______; 99,94; 99,83; 99,72;... complete the table below. Input number | 1 | 2 | 3 | 4 | 5 | 12 | n | ||

Input number + 7 | 8 | 11 | 15 | 30 |

### Multiplying or separating with the same number

Take an additional look at sequence F: 2; 6; 18; 54; 162; 486; ...

Piet describes that he figured out how to proceed the succession F:

I looked at the very first two terms in the sequence and wrote (2 imes ? = 6).

When ns multiplied the an initial number by 3, I gained the second number: (2 imes 3 = 6).

I then checked to see if I could find the following number if i multiplied 6 by 3: (6 imes 3 = 18).

I ongoing checking in this way: ( 18 imes 3 = 54; 54 imes 3 = 162) and so on.

This offered me **a rule I deserve to use to prolong the sequence** and my dominion was: **multiply each number by 3 to calculate the next number in the sequence.**

Zinhle says you can likewise find the pattern by working backwards and also dividing by 3 each time:

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

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

check whether Piet"s thinking works because that sequence H: 2; 4; 8; 16; 32; 64; ... Describe, in words, the dominance for detect the next number in the sequence. Likewise write down the next five terms that the succession if the sample 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 absent output and also input numbers:

What is the term-to-term dominance for the output numbers here, (+ 6 ext or imes 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 answer the questions 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 the the exact same kind as sequence B? Explain. In what way are sequences B and also E different from the other sequences?There space sequences wherein there is neither a consistent difference nor a continuous ratio in between consecutive terms and also yet a sample still exists, as in the instance of order B and E.

take into consideration the sequence: 10; 17; 26; 37; 50; ... create down the next 5 numbers in the sequence. Eric observed the he deserve to calculate the following term in the sequence as follows: 10 +

**7**= 17; 17 +

**9**= 26; 26 +

**11**= 37. Usage Eric"s technique to inspect whether your numbers in concern (a) over are correct. which of the statements below can Eric use to describe the relationship between the numbers in the succession in question 2? check the dominion for the very first three terms of the sequence and also then simply write "yes" or "no" beside each statement. rise the difference between consecutive state by 2 every time rise the difference between consecutive terms by 1 each time include two much more than you added to gain the previous ax administer a preeminence to describe the relationship in between the number in the sequences below. Usage your rule to provide 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 connection in a sequence

### Using place to do predictions

Take an additional look in ~ equences A to H. Which sequence(s) are of the same kind as 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 been thinking about Amanda and also Tamara"s explanations of just how they settled the ascendancy for succession A and also has drawn up a table. The agrees through them yet says that there is one more rule that will also work. That explains:

My table shows the terms in the sequence and also the difference 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 rule will likewise work:

**Multiply the place of the number by 3 and include 2 come the answer.**

I have the right to write this preeminence as a number sentence: **Position that the number(f imes 3 + 2)**

I usage my number sentence to check: ( f1 imes 3 + 2 = 5; f2 imes 3 + 2 = 8; f3 imes 3 + 2 = 11 )

What execute the number in bolder in Sizwe"s number sentence stand for? What walk the number 3 in Sizwe"s number sentence was standing for? think about the sequence 5; 8; 11; 14; ...Apply Sizwe"s dominion to the sequence and also determine:

hatchet number 7 of the sequence hatchet number 10 of the sequence the 100th hatchet of the succession think about the sequence: 3; 5; 7; 9; 11; 13; 15; 17; 19;.. use Sizwe"s explanation to find a preeminence for this sequence. identify the 28th hatchet of the sequence.### More predictions

Complete the tables listed below by calculating the missing 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 |

**Position in the sequence ( imes) (position in the succession + 1)**to finish the table below.

Position in sequence | 1 | 2 | 3 | 4 | 5 | 6 |

Term | 2 |

## Investigating and extending geometric patterns

### Square numbers

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

### Triangular numbers

Therese offers circles to kind a sample of triangle shapes:

If the pattern is continued, how numerous circles have to Therese have in the bottom heat of picture 5? in the second row from the bottom of snapshot 5? in the 3rd row indigenous the bottom of photo 5? in the second row from the optimal of snapshot 5? in the top row of photo 5? in total in picture 5? display your calculation. How plenty of circles go Therese need to kind triangle picture 7? display the calculation. How plenty of circles walk Therese need to form triangle photo 8? complete the table below. Display all her work.

Picture number | 1 | 2 | 3 | 4 | 5 | 6 | 12 | 15 |

Number that circles | 1 | 3 | 6 | 10 |

More 보다 2 500 year ago, Greek mathematicians currently knew that the numbers 3, 6, 10, 15 and also so top top could type a triangular pattern. They stood for these numbers with dots which they i ordered it in such a means that they developed equilateral triangles, therefore the surname **triangular numbers**. Algebraically us think of them as sums of consecutive herbal numbers beginning with 1.

Let us revisit the task on triangular numbers that us did in the ahead section.

So far, we have identified the number of circles in the sample by adding consecutive natural numbers. If we were request to recognize the number of circles in snapshot 200, because that example, it would take united state a an extremely long time to carry out so. We require to discover a quicker method of finding any type of triangular number in the sequence.

Consider the plan below.

We have included the yellow circles to the original blue circles and also then rearranged the one in together a means that they are in a rectangle-shaped form.

photo 2 is 3 circles long and 2 one wide. Finish the following sentences: photo 3 is ______ circles long and also ______ circles wide. photo 1 is ______ circles long and ______ circle wide. snapshot 4 is ______ circles long and ______ one wide. picture 5 is ______ circles long and ______ circles wide. How countless circles will there be in a photo that is: 10 circles long and also 9 circles wide? 7 one long and 6 circles wide? 6 circles long and also 5 one wide? 20 circles long and 19 one wide?Suppose we want to have actually a quicker an approach to recognize the number of circles in picture 15. We understand that snapshot 15 is 16 one long and also 15 circles wide. This gives a complete of (15 imes 16 = 240) circles. But we must compensate for the fact that the yellow one were initially not over there by halving the total variety of circles. In other words, the original number has (240 div 2 = 120) circles.

usage the above reasoning to calculation the number of circles in: picture 20 snapshot 35## Describing patterns in various ways

### T-shaped numbers

The pattern listed below is make from squares.

How numerous squares will there it is in in pattern 5? How numerous squares will certainly there be in sample 15? complete the table.

Pattern number | 1 | 2 | 3 | 4 | 5 | 6 | 20 |

Number that squares | 1 | 4 | 7 | 10 |

Below space three various methods or plan to calculation the variety of squares because that pattern 20. Study each one carefully.

**Plan A:**

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

**Plan B:**

Multiply the sample number by 3, and subtract 2. So pattern 20 will have actually (20 imes 3 - 2) squares.

**Plan C:**

The number of squares in sample 5 is 13. So sample 20 will have actually (13 imes 4 = 52) squares due to the fact that (20 = 5 imes 4).

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### ... And some other shapes

Three numbers are offered below. Draw the next figure in the tile pattern. If the pattern is continued, how many tiles will certainly there it is in in the 17th figure? answer this question by analysing what happens. Thato decides the it much easier for the to watch the pattern as soon as the tiles room rearranged as displayed here:Use Thato"s technique to determine the number of tiles in the 23rd figure.

finish the flow diagram listed below by creating the suitable operators so the it deserve to be provided to calculate the variety of tiles in any type of figure the the pattern. How numerous tiles will certainly there be in the 50th number if the sample is continued?write down the next 4 terms in every sequence. Also explain, in every case, how you figured out 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 |

How numerous cubes will certainly there it is in in stack 5? complete the table.

Stack number

1 | 2 | 3 | 4 | 5 | 6 | 10 |

Number the cubes | 1 | 8 | 27 |