Transcript 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 sequences
For 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?
Patterns and sequences
For any pattern it is important to try to spot
what is happening before you can predict the
next number.
1, 2, 4,…
Who thought that the next
number was 3?
What comes next?
Patterns and sequences
For any pattern it is important to try to spot
what is happening before you can predict the
next number.
1, 2, 4, 8, 16, …
What comes next?
Patterns and sequences
Look at what is happening from 1 TERM to
the next. See if that is what is happening
for every TERM.
5,
+3
8,
12,
17,
23,
…,
…
Patterns and sequences
Look at what is happening from 1 TERM to
the next. See if that is what is happening
for every TERM.
5,
+3
8,
+3
X
12,
17,
23,
…,
…
Patterns and sequences
Look at what is happening from 1 TERM to
the next. See if that is what is happening
for every TERM.
5,
+3
8,
+4
12,
17,
23,
…,
…
Patterns and sequences
Look at what is happening from 1 TERM to
the next. See if that is what is happening
for every TERM.
5,
+3
8,
+4
12,
+5

17,
23,
…,
…
Patterns and sequences
Look at what is happening from 1 TERM to
the next. See if that is what is happening
for every TERM.
5,
+3
8,
+4
12,
+5
17,
+6

23,
…,
…
Patterns and sequences
Look at what is happening from 1 TERM to
the next. See if that is what is happening
for every TERM.
5,
+3
8,
+4
12,
+5
17,
+6
23,
+7
30,
…
Patterns and sequences
Now try these patterns:
3,
7,
11,
15,
19,
…,
…
128,
64,
32,
16,
8,
…,
…
10,
1,
…,
…
135,
…,
…
1000,
5,
100,
15,
45,
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
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
• 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
Finding the nth term of an arithmetic sequence
Tn = 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, 16
Tn = 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 constant
Find the difference of the difference
4
4
As 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 T2
Tn = an² + bn + c
T2 = 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