Linear SequencesLinear Sequences

Slideshow 7, Room 307Mr Richard Sasaki, Mathematics

Slideshow 7, Room 307Mr Richard Sasaki, Mathematics

• Find patterns in sequences• Make formulae for sequences (the n-

th term)• Use formulae to find positions of

certain numbers

Objectives

First, please try the 5 minute vocabulary check. The purpose of this is to not fall for decoy information, good luck!

Answers5

5

240 Yen

5

7

4

3

15 (or 12)

1050ml

5 (naan bread and tandoori chicken)

SequencesA sequence is an ordered number pattern. It is often easy to see which numbers are missing in the pattern or the next numbers that come.

3, 5, 7, 9, __, 13, __, __11 15 17

Here it was easy to tell that the numbers increase by 2 every step to the right.If numbers go up (or down) in the same way every step, then the sequence is linear.

SequencesWith sequences, it is important to understand each number’s position.

3, 5, 7, 9, __, 13, __, __11 15 17

1 2 3 4 5 6 7 8

We call the position .So for the second position (where ), we have 5.How about the 20th position ()?Well done! But what calculation did we do to get to 41?

20

41

Position

Number

SequencesThe formula must be “in terms of” . This means that the formula must be about .

3, 5, 7, 9, __, 13, __, __11 15 17

1 2 3 4 5 6 7 8 20

41

Position

Number

Let’s try to make a formula for this sequence. 𝑛

The formula must contain the unknown as we relate it to each number’s position.The formula goes up in twos. So we need to multiply the unknown by 2.

2 Is that it?Let’s check.

( means .)

SequencesIf the formula is , we multiply (the position) by .

𝑛2 3, 5, 7, 9, __, 13, __, __11 15 17

1 2 3 4 5 6 7 8Position

Number

2𝑛 2,4, 6,8,10,12,14, 16All of the numbers in our test are slightly off, how much by?

+1

We need to add 1 to each.Example

Find a formula in terms of for the sequence below.7, 10, 13, 16, 19, …

Finished!

The numbers increase by each time.3𝑛3…?

Sequences

7, 10, 13, 16, 19, …

1 2 3 4 5Position

Number𝑛3…?

A quicker way to do this is to find the 0th term. 0

__,4

We simply add the 0th term to our formula.

+4

Using , we can reproduce our sequence to check if it is correct.

3𝑛+4𝑛=17,𝑛=2

10,𝑛=3

13,It looks good!

Sequences

3

ExampleFind a formula for the nth term for the sequence below. Also, find out what the 50th term is.

2, 7, 12, 17, 22, …How much do the numbers increase by?

5What would the 0th term be?

-3,

-3What is the formula for the nth term? -𝑛5

What would the 50th term be?5𝑛 –3 ,𝑛=50

(5×50)−3

247

¿247

Answers - Easy8 1214

1 4 1950 6266

21 3339

-5-3 3

2 -10 -14

2

58

0.5

-327

1

0

-5

1

-4

−1372𝑛+2

Answers - Medium32

3𝑛+2

2𝑛+15𝑛8𝑛−50.5𝑛+1

−3𝑛−42𝑛7−137

102228406𝑛−2

6×10−2=586×100−2=5985 ,7 ,9 ,11 ,132×50+3=103

Answers - Hard

612 16

6 9

4.55.5

-1-2

-56

74

6 𝑎

1, 5, 9, 13, 17, 21, 25, 29

4×35−3=137, …5811

2𝑛𝑛+13𝑛+1

𝑛+32𝑛+83𝑛−3𝑛+2.5−𝑛+211𝑛−16−3𝑛+132𝑎𝑛

3𝑛+2