Sequences & Series ♥

By Jen, Cindy, Tommy & Samuel

The cool people of group e ☻

6.1 sequences

SequencesA sequence is a set of numbers in a particular order where each number is derived from a particular rule

We have done sequences before. Remember the “nth” term?

BUT let’s refresh our memory ☺5 7 9 11……

Can you find the 10th term?

Can you find the nth term?

Which term gives the number 63?

When we mention the nth term, we write it as un

It can be called the general term of the sequence

Let’s check out a quick question (its on pg 175 on the IB book)

un = n² + 6n for n ≥ 1

If n = 6 , it should be written as u6

So the answer for when n = 6 is:

u6 = (6)² + 6(6) = 72

General term of the sequence

Recurrence relation

• A recurrence relation is when relationship between two terms which occurs throughout the sequence.

• Example: un+1 = 2un + 1• As you can see, the general term

of the sequence is in both sides of the equation.

How to check your answer on you GDC…!

• If you were asked to find the first 3 terms of the sequence , find the first term of the sequence first and input it into your GDC .

• The first term is• You find the answer is 3.• Input the equation onto your GDC , for

example, ‘2 x AnS +1’ and then keep pressing enter to check

12 nun

1)1(21 u

Quick questions

• Write down the first 4 terms of these sequences- Check using the GDC

• Write down an expression in terms of n for the nth term of this sequence…

12 nun )1(

1

nnun

...12

1,

9

1,

6

1,

3

1

6.2 Series & Sigma notation

Series and Sigma Notation.

• A SERIES is the sum of the terms in a sequence. You write the sum of the first n terms of the sequence as . This represents…

• This is an example of FINITE SERIES because there is a finite number of terms. This means there is a specific number of terms and the sequence doesn’t go on forever. It can be expressed in more detail using sigma notation, which we have learnt.

nn uuuuS .....321

nS

n

r

rn uuuuu1

321 ...This means “the sum of the outcomes of the sequence from r=1 to n

ru

ru

Continued…

• For example, the sum of 7, 11, 15 and 19 is shown by…

• The infinite series of 3+5+7+9+… can be written as……

• REMEMBER ‘+…’ means the series goes on and on…

4

1

)34(r

r

1

)12(r

r

6.3 Arithmetic Sequences or

Arithmetic Progression

The basicsAn arithmetic sequence is a sequence of numbers

such that the difference of any 2 successive members of the sequence is constant.

That basically meant that the difference between one member and the next stays the same.

Eg. 1, 3, 5, 7, 9, 11 …For this arithmetic sequence, the difference is 2!!!

Moving on…

• There is an all-knowing, all-doing formula for arithmetic sequences.

• It can also be written as:

REMEMBER IT!!!

Proof to Sum of AP

6.4 & 6.5 Geometric Sequences & Infinite

geometric sequences

Previously we worked mainly with sequences made by multiplying constants in the equation

eg. 3n + 4

Geometric sequences have the n as a power of a constant

eg. 4 x 3(n-1)

Geometric Sequences

Comparison between x2 and 2x sequences

y x2 2x

0 0 1

1 1 2

2 4 4

3 9 8

4 16 16

5 25 32

6 36 64

7 49 128

y = 1

y = x

y = x2

y = 2x

Geometric Sequences

u1rn-1

You need 3 variables to form a geometric sequence

u1 and r are constants in the term

n is for finding the n-th term

r is called the common ratio

y = 2x

Graph for yeast population growth

Another way to look at geometric sequences

y = 21 y = 22 y = 23

y = 20

I’m sure you can figure out what y = 24 would be like …

example sequence:2, 6, 18, 54, 162, 486 …

by which u1 = 2 , r = 3

so by putting it in into the u1rn-1

2 x 3(n-1)

2 x 30 2 x 1 22 x 31 2 x 3 62 x 32 2 x 9 182 x 33 2 x 27 542 x 34 2 x 81 1622 x 35 2 x 243 486

• Such sequences can also get smaller

• consider u1 = 1, r = ½

• ½(n-1)

½0 1

½1 0.5

½2 0.25

½3 0.125

½4 0.0625

Which leads to 2

11/2

1/41/8

1/161/32

Another Example

• You are learning a new topic in a difficult subject. On the first day of learning the topic, your teacher gives you 64 questions to work on (on the subject)

• On the second day, you are given 32 questions, then on the third day, you are given 16 questions.

• On the 4th day you would get 8 and on the fifth day you would get 4 questions.

• This is done by the equation

• 64 (½(x-1))• Which would give

the following answers table

x y

0 128

1 64

2 32

3 16

4 8

5 4

6 2

y = 64 (½(x-1))

Sum of a geometric sequence

u1rn-1

if u1 = 1 and r = 3

and you have find the sum of the first 8 terms

S8 = 1 + 3 + 9 + 27 + … + 729 + 2187

3S8 = 3 + 9 + 27 + 81 … + 2187 + 6561

3S8 – S8 = -1 + 6561

2S8 = 6560

S8 = 3280

• Now, question time…

Infinite Geometric Sequences

• Now, question time…(again)

6.6 Binomial Expansions

Binomial Expansions

• A binomial is an expression which contains 2 terms.• E.g.: (a+b), (1+x), (5+y)• Binomial expansion is basically looking at “opening the brackets”

of these binomials, which we have looked at before.• Take a look…

(1+x) = 1+x(1+x)² = 1 + 2x + x²

(1+x)³ = (1+x)²(1+x) = (1 + 2x + x²)(1+x) = 1 + 3x + 3x² + x³ • This is how you expand each one.

Can you spot the pattern?

• Let’s list out some of the expansions of (1+x)n

(1 + x) = 1+x(1 + x)² = 1 + 2x + x²(1 + x)³ = 1+ 3x + 3x²+ x³(1 + x)4 = 1 + 4x + 6x2 + 4x3 + x4

(1 + x)5 = 1 + 5x + 10x2 + 10x3 + 5x4 + x5

• Hmm… can you see something there???What if I list out the coefficients?

(1 + x) = 1+1(1 + x)² = 1 + 2 + 1(1 + x)³ = 1+ 3 + 3+ 1(1 + x)4 = 1 + 4 + 6 + 4 + 1(1 + x)5 = 1 + 5 + 10 + 10 + 5 + 1

Pascal’s Theorem & Binomials

•As we all know, the next row is obtained like so:

1 4 6 4 1

1 (1+4) (4+6) (6+4) (4+1) 1

1 5 10 10 5 1

1 5x 10x2 10x3 5x4 x5

•The coefficient of the x² term in the expansion of (1+x)5 is 10.

•The coefficient of the x4 term in the expansion of (1+x)5 is 5

• Quick Question – what is the coefficient of the x9 term in the expansion of (1+x)10?

General expressions

• The general expression of the (r + 1)th term in the expansion of (1 + x)n is:

= remember, “!” is a FACTORIAL!

• You can use your GDCs to work out the value of and if you look now, it is usually written as nCr or nCr.

• Using the results, we can now get a general formula for the expansion of (1+x)n

(1+x)n = 1 + x + x2 + x3 +………xn

HURRAY XD

)!(!

!

rnr

n

r

n

r

n

1

n

2

n

3

n

Expansions of (a+x)n

• As with any binomial equation, the expansions are linked with Pascal's theorem.

Can you figure out:(a + x)1 =(a + x)2 =(a + x)3 =

Using the previous general rule for the expansion of (1+x)n, we can figure out a general formula for the expansion of (a+x)n , which is usually known as the binomial theorem.

• (a+x)n = an + an-1x + an-2x2 + an-3x3 +…….+xn

• We can also use this equation for approximations, which we will see in the book.

1

n

2

n

3

n

Questions =]

• Expand: (2x-y)5

• Find the coefficient of the term x4 in the expansion of the equation (1+x)7

MORE Questions =]

• WITHOUT using a calculator…

Simplify ( + )6 + ( - )67 3 7 3

The End =]