patterns and sequences

Patterns and sequences

We often need to spot a pattern in order to predict what will happen next.

In maths, the correct name for a pattern of numbers is called a SEQUENCE.

The first number in a SEQUENCE is called the FIRST TERM T1; the second is the SECOND TERM T2, the nth term is Tn, and so on.

Patterns and sequences

For any pattern it is important to try to spot what is happening before you can predict the next number.

The first 2 or 3 numbers is rarely enough to show the full pattern - 4 or 5 numbers are best.

Patterns and sequencesFor any pattern it is important to try to spot what is happening before you can predict the next number.

1, 2, …… What’s the next number?


1, 2, 4,… Who thought that the next

number was 3?

What comes next?


1, 2, 4, 8, 16, …

What comes next?

Patterns and sequencesLook at what is happening from 1 TERM to the next. See if that is what is happening for every TERM.

5, 8, 12, 17, 23, …, …

+ 3


5, 8, 12, 17, 23, …, …

+ 3 + 3

X


5, 8, 12, 17, 23, …, …

+ 3 + 4


5, 8, 12, 17, 23, …, …

+ 3 + 4

+ 5


5, 8, 12, 17, 23, …, …

+ 3 + 4

+ 5

+ 6


5, 8, 12, 17, 23, 30, …

+ 3 + 4

+ 5

+ 6

+ 7

Patterns and sequencesNow try these patterns:

3, 7, 11, 15, 19, …, …

128, 64, 32, 16, 8, …, …

1000, 100, 10, 1, …, …

5, 15, 45, 135, …, …

Arithmetic Sequence

• A sequence, in which any term after the first can be obtained by adding a fixed number to the term before it, is called an arithmetic sequence.

Term 1 ( T1) = a

The fixed number is called the common difference and is represented by the letter d.

T1 (a) = 2

d = 2

The nth term of an arithmetic sequence

• Example : If the nth term of a sequence is 4n – 3, write down the first 5 terms of the sequence.

• Tn = 4n -3• T1 = 4(1) -3 = 1• T2 = 4(2) -3 = 5• T3 = 4(3) – 3 =9• T4 = 4(4) – 3 = 13• T5 = 4(5) – 3 = 17

When using a number sequence we sometimes need to know, for example, the 50th or 1000th term. To do this we need to find a rule that generates the sequence

Finding the nth term of an arithmetic sequenceTn = a + ( n -1)d

In an arithmetic sequence 3, 8, 13,…….Find (i) a (ii) d (iii) Tn (iv) T20

(i) a = T1 = 3(ii) d = common difference = 5(iii) Tn = a + (n -1)d = 3 + (n – 1)5 = 3 + 5n – 5 = 5n – 2

(iv) T20 = 5(20) – 2 = 100 – 2 = 98

(i) Find the nth term of the arithmetic sequence 7, 10, 13, 16Tn = a + ( n – 1) d 7 + ( n – 1 ) 3 7 + 3n – 3 3n + 4

(ii) Which term of the sequence is 97 ? 3n + 4 = 97 3n = 93 n = 31

(iii) Show that 168 is not a term of the sequence. 3n + 4 = 168 3n = 164 n = 164 / 3, since n is not a whole number, 168 is not a term in

the sequence.

Finding the values of a and d

If we are given any two terms of an arithmetic sequence, we can use simultaneous equations to find the value of a and d.

Example: T4 of an arithmetic sequence is 11 and T9 = 21

Find the values of a and d and hence find T50.

Tn = a + (n – 1)d

T4 : a + 3d = 11

T9 : a + 8d = 21

By using simultaneous equations, a = 5 and d = 2

Arithmetic Series

• The African-American tribal celebration involves the lighting of candles every night for seven nights. The first night one candle is lit and blown out.

Arithmetic Series

• The second night a new candle and the candle from the first night are lit and blown out. The third night a new candle and the two candles from the second night are lit and blown out.

Arithmetic Series

• This process continues for the seven nights.

• We want to know the total number of lightings during the seven nights of celebration.

Arithmetic Series• The first night one candle was

lit, the 2nd night two candles were lit, the 3rd night 3 candles were lit, etc.

• So to find the total number of lightings we would add:

• 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28

Arithmetic Series

• Arithmetic sequence: 2, 4, 6, 8, 10

• Corresponding arithmetic. series:2 + 4 + 6

+ 8+10

• Arithetic Sequence: -8, -3, 2, 7

• Corresponding Arithmetic Series: -8 + -3 + 2 + 7

Arithmetic Series

• Sn is the symbol used to represent the first ‘n’ terms of a series.

• Given the sequence 1, 11, 21, 31, 41, 51, 61, 71, … find S4

• We simply add the first four terms 1 + 11 + 21 + 31 = 64

Arithmetic Series• Find S8 of the arithmetic sequence

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 =36

Arithmetic Series

What if we wanted to find S100 for the sequence in the last example. It would be very long to have to list all the terms and try to add them up.

• Let’s figure out a formula!! :)

Arithmetic Series

When the terms of an arithmetic sequence are added, they form an arithmetic series.

Example 1,3,5,7 …….is an arithmetic sequence

1+3+5+7 ….is an arithmetic series.

We use Sn to show the sum of the first n terms.

We use the following formula to find the sum of n terms of an arithmetic series.

Quadratic Sequences

A sequence is a set of numbers that are connected in some way. In this section we will look at quadratic sequences where the difference between the terms changes.

Consider the following sequence :

Here, the differences between terms are not constant, but there is still a pattern.-the differences between each number in the sequence vary-But the second difference is a constant 2.When the second difference is constant, you have a quadratic sequence - ie, there is an n2 term.

Quadratic Sequences

Learn these rules:

If the second difference is 2, you start with n2.

If the second difference is 4, you start with 2n2.

If the second difference is 6, you start with 3n2.

Finding the nth term of a quadratic sequence

Consider the sequence 3, 10, 21, 36 Work out each difference 7 11 15 diff is not constantFind the difference of the difference 4 4As the difference of the difference is constant, it is a quadratic sequence

Tn = an² + bn + c……. Now write T1 and T2 in terms of a and b T1 = 2(1)² + b(1) + c…..a = 2 because it is half the second difference 2 + b + c = 3 ………because we know from the sequence that T1 = 3 b + c = 3 – 2 b + c = 1 ( This is equation 1)

Now do the same for T2Tn = an² + bn + cT2 = 2(2) ² + b(2) + c = 8 + 2b + c = 10……..because we know from the sequence T2 = 10 = 2b + c = 2 ……….. ( This is equation 2 )

You now solve for b and c, by means of Simultaneous Equations.B = 1 anc c = 0

patterns and sequences

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