Sequences Revision

Learning Objective:Learning Objective:Arithmetic SequencesArithmetic SequencesGeometric SequencesGeometric SequencesNth termsNth termsSumsSums

Arithmetic SequencesAn arithmetic sequence has an recurrence relationship of the form:

u1 = a number un+1 = un + d

The number added on each time is called the common difference.

Arithmetic SequencesWhich of these are arithmetic sequences?

un+1 = un - 4

un+1 = un + 11

un+1 = 11un - 4

un+1 = 2un - 4

un+1 = (un)2

Geometric SequencesA geometric sequence has an inductive definition of the form:

u1 = a un+1 = r un

The number multiplied by each time is called the common ratio.

Geometric SequencesThe sequence 3, 12, 48, 192, 768 can be defined by…

u1 = 3 un+1 = 4 un

u2 = 3 x 4 u2 = 12

u3 = 3 x 4 x 4 u3 = 48

u4 = 3 x 4 x 4 x 4 u4 = 192

un = 3 x 4n-1

1st term Common ratio

Sequences For the sequence :

6, 11, 16, ……, 731

How many terms?

un = u1 + (n-1)d

731 = 6 + 5(n-1)

u1 = d = 56 un = 731

n -1 = (731-6)/5 = 145 n = 146

What sort of question?What sort of question? nth term of arithmetric sequencenth term of arithmetric sequence

Sequences If I put £400 in a bank account on my 16th birthday and get 5% interest per year.

How much money would I have on my 41st birthday?

un = u1 x r n-1

un = 400 x 1.05 24

u1 = r = 1.05400 n = 25

un = £1290.04

What sort of question?What sort of question? nth term of geometric sequencenth term of geometric sequence

Sequences For the sequence:

7, 9, 11, 13, …..

What is the sum of the first 50 terms?

What sort of question?What sort of question? Sum of terms in arithmetic sequenceSum of terms in arithmetic sequence

Sn = n/2 (2u1 + (n-1)d)

S50 = 25(14 + 49 x 2)

u1 = d = 27 n =

S50 = 25 x 112 = 2800

50

Sequences The sequence:

2, 4a, 8a2, ….

Find an expression for the sum of the first 10 terms

S10 = u1 (r n – 1)/(r-1)

S10 = 2((2a)10-1)/(2a-1)

u1 = r = 2a2 n = 10

S10 = (211a10-2)/(2a-1)

What sort of question?What sort of question? sum terms of geometric sequencesum terms of geometric sequence

The 1The 1stst term of a geometric sequence is 12 and the sum to Infinity is 9. term of a geometric sequence is 12 and the sum to Infinity is 9.

Find the Find the ratio of termsratio of terms

u1 = 12

91

12

r

12 = 9(1-r) 12 = 9 - 9r

9r = 9 - 12 = -3

r = - 1/3

Converging series

A geometric series converges to a limit when..A geometric series converges to a limit when..

1r

Means the size of r including negatives

Converging seriesWhich of the following values of r will converge to Which of the following values of r will converge to a limit?a limit?

0.9

0.3

0.005

-0.2

-1.5

-1-0.03

1r

Example 3

2

211

21

1

1

S

r =

Evaluate:

0 2

1

n

n

1/2

1st term, when n=0 u1 = (1/2)0 = 1

r

uS

1

1

Different sorts of question to try

1 3

1

n

n

.....008.008.08.08.0.

As a fraction

Sum to infinity

1 + 2/5 + 4/25 + … r =

r =

r =

u1 =

u1 =

u1 =

r

uS

1

1

2/5

1/3

1/10

1

(1/3)1=1/3

0.8= 4/5

sum = 1 / 3/5 = 5/3

sum = 1/3 / 2/3 = 1/2

sum = 4/5 / 9/10 = 8/9