Sequences and Series

Mathematics 4

February 1, 2012

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Infinite Sequences

An infinite sequence (or simply sequence) is a function whose domainis the set of positive integers.

Example: an = 3n+ 1

term number 1 2 3 4 5 n

term value 4 7 10 13 16 3n+ 1

This sequence is said to be explicitly defined.

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Infinite Sequences

An infinite sequence (or simply sequence) is a function whose domainis the set of positive integers.

Example: an = 3n+ 1

term number 1 2 3 4 5 n

term value 4 7 10 13 16 3n+ 1

This sequence is said to be explicitly defined.

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Infinite Sequences

An infinite sequence (or simply sequence) is a function whose domainis the set of positive integers.

Example: an = 3n+ 1

term number 1 2 3 4 5 n

term value 4 7 10 13 16 3n+ 1

This sequence is said to be explicitly defined.

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Explicitly Defined Infinite Sequences

List the first three terms and the tenth term of each sequence

1. an =n

n+ 1

12 ,

23 ,

34 , a10 =

1011

2. an = (−1)n+1 n2

3n− 112 .−

45 ,

98 , a10 = −

10029

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Explicitly Defined Infinite Sequences

List the first three terms and the tenth term of each sequence

1. an =n

n+ 112 ,

23 ,

34 , a10 =

1011

2. an = (−1)n+1 n2

3n− 112 .−

45 ,

98 , a10 = −

10029

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Explicitly Defined Infinite Sequences

List the first three terms and the tenth term of each sequence

1. an =n

n+ 112 ,

23 ,

34 , a10 =

1011

2. an = (−1)n+1 n2

3n− 1

12 .−

45 ,

98 , a10 = −

10029

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Explicitly Defined Infinite Sequences

List the first three terms and the tenth term of each sequence

1. an =n

n+ 112 ,

23 ,

34 , a10 =

1011

2. an = (−1)n+1 n2

3n− 112 .−

45 ,

98 , a10 = −

10029

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Recursively Defined Infinite Sequences

A recursively defined sequence is a sequence where the first term a1 isstated, together with a rule for obtaining the next term an+1 from thepreceding term an.

List the first five terms of each sequence

1. a1 = 1, an+1 = 7− 2an

1, 5,−3, 13,−19, 452. a1 = a2 = 1, an = an−1 + an−2

1, 1, 2, 3, 5

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Recursively Defined Infinite Sequences

A recursively defined sequence is a sequence where the first term a1 isstated, together with a rule for obtaining the next term an+1 from thepreceding term an.

List the first five terms of each sequence

1. a1 = 1, an+1 = 7− 2an

1, 5,−3, 13,−19, 452. a1 = a2 = 1, an = an−1 + an−2

1, 1, 2, 3, 5

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Recursively Defined Infinite Sequences

A recursively defined sequence is a sequence where the first term a1 isstated, together with a rule for obtaining the next term an+1 from thepreceding term an.

List the first five terms of each sequence

1. a1 = 1, an+1 = 7− 2an

1, 5,−3, 13,−19, 45

2. a1 = a2 = 1, an = an−1 + an−2

1, 1, 2, 3, 5

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Recursively Defined Infinite Sequences

A recursively defined sequence is a sequence where the first term a1 isstated, together with a rule for obtaining the next term an+1 from thepreceding term an.

List the first five terms of each sequence

1. a1 = 1, an+1 = 7− 2an

1, 5,−3, 13,−19, 452. a1 = a2 = 1, an = an−1 + an−2

1, 1, 2, 3, 5

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Recursively Defined Infinite Sequences

A recursively defined sequence is a sequence where the first term a1 isstated, together with a rule for obtaining the next term an+1 from thepreceding term an.

List the first five terms of each sequence

1. a1 = 1, an+1 = 7− 2an

1, 5,−3, 13,−19, 452. a1 = a2 = 1, an = an−1 + an−2

1, 1, 2, 3, 5

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Infinite Sequences

Examples

1. The number of bacteria in a certain culture is initially 200, and theculture doubles in size every hour. Find an explicit and a recursiveformula for the number of bacteria present after n hours.

2. Use a calculator to determine the value of a4 in the sequencea1 = 3, an+1 = an − tan an.

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Infinite Sequences

Examples

1. The number of bacteria in a certain culture is initially 200, and theculture doubles in size every hour. Find an explicit and a recursiveformula for the number of bacteria present after n hours.

2. Use a calculator to determine the value of a4 in the sequencea1 = 3, an+1 = an − tan an.

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Arithmetic Sequences

A sequence a1, a2, a3, ..., an, ... is an arithmetic sequence if each termafter the first is obtained by adding the same fixed number d to thepreceding term.

an+1 = an + d

The number d = an+1 − an is called the common difference of thesequence.

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Arithmetic Sequences

Given the diagram below:

1. Determine the common difference between diagrams.

2. How many blocks will Diagram 10 have?

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Arithmetic SequencesFinding the nth term of an AS

The nth term of an arithmetic sequence is given by:

an = a1 + (n− 1)d

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Arithmetic Sequencesan = a1 + (n− 1)d

1. Find the 25th term of the arithmetic sequence 2, 5, 8, 11, ...

2. The 18th and 52nd terms of an AS are 3 and 173 respectively. Findthe 25th term.

3. The terms between any two terms of an arithmetic sequence arecalled the arithmetic means between these two terms. Insert fourarithmetic means between -1 and 14.

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Arithmetic Sequencesan = a1 + (n− 1)d

1. Find the 25th term of the arithmetic sequence 2, 5, 8, 11, ...

2. The 18th and 52nd terms of an AS are 3 and 173 respectively. Findthe 25th term.

3. The terms between any two terms of an arithmetic sequence arecalled the arithmetic means between these two terms. Insert fourarithmetic means between -1 and 14.

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Arithmetic Sequencesan = a1 + (n− 1)d

1. Find the 25th term of the arithmetic sequence 2, 5, 8, 11, ...

2. The 18th and 52nd terms of an AS are 3 and 173 respectively. Findthe 25th term.

3. The terms between any two terms of an arithmetic sequence arecalled the arithmetic means between these two terms. Insert fourarithmetic means between -1 and 14.

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Arithmetic SequencesPartial Sum of an Arithmetic Sequence

The sum of the first n terms of an arithmetic sequence is given by theformula:

Sn =n[2a1 + (n− 1)d]

2

or

Sn =n(a1 + an)

2

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Arithmetic Sequences

Sn =n(a1 + an)

2=

n[2a1 + (n− 1)d]

2

1. Find the sum of the first 30 terms of the arithmetic sequence -15,-13, -11, ...

2. The sum of the first 15 terms of an arithmetic sequence is 270.Find a1 and d if a15 = 39.

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Arithmetic Sequences

Sn =n(a1 + an)

2=

n[2a1 + (n− 1)d]

2

1. Find the sum of the first 30 terms of the arithmetic sequence -15,-13, -11, ...

2. The sum of the first 15 terms of an arithmetic sequence is 270.Find a1 and d if a15 = 39.

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Harmonic Sequences

A harmonic sequence is a sequence of numbers whose reciprocalsform an arithmetic sequence.

1. Find two harmonic means between 4 and 8.

2. Find the 14th term of the harmonic sequence starting with 3, 1.

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Harmonic Sequences

A harmonic sequence is a sequence of numbers whose reciprocalsform an arithmetic sequence.

1. Find two harmonic means between 4 and 8.

2. Find the 14th term of the harmonic sequence starting with 3, 1.

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Homework 6

Vance p. 179 numbers 2, 4, 6, 10, 12, 14, 15, 18, 24, 25.

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Geometric Sequences

A geometric sequence is a sequence in which each term after the firstis obtained by multiplying the same fixed number, called the commonratio, by the preceding term.

gn+1 = gn · r

The number r =gn+1

gnis called the common ratio of the sequence.

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Geometric Sequences

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Geometric Sequencesgn+1 = gn · r

Examples:

1. Give the next 3 terms of the GS 27, 9, 3, ...

1, 13 ,19

2. Find the 10th term of the GS -8, 4, -2, ...

164

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Geometric Sequencesgn+1 = gn · r

Examples:

1. Give the next 3 terms of the GS 27, 9, 3, ... 1, 13 ,19

2. Find the 10th term of the GS -8, 4, -2, ...

164

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Geometric Sequencesgn+1 = gn · r

Examples:

1. Give the next 3 terms of the GS 27, 9, 3, ... 1, 13 ,19

2. Find the 10th term of the GS -8, 4, -2, ...

164

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Geometric Sequencesgn+1 = gn · r

Examples:

1. Give the next 3 terms of the GS 27, 9, 3, ... 1, 13 ,19

2. Find the 10th term of the GS -8, 4, -2, ... 164

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Geometric Sequencesgn = g1 · rn−1

A geometric sequence can also be expressed explicitly as:

gn = g1 · rn−1

Examples

1. If the 8th term of a GS is 243 and the 5th term is 9, write the first3 terms.

19 ,

13 , 1

2. Find the 1st term of a GS with g5 = 162 and r = −3.

2

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Geometric Sequencesgn = g1 · rn−1

A geometric sequence can also be expressed explicitly as:

gn = g1 · rn−1

Examples

1. If the 8th term of a GS is 243 and the 5th term is 9, write the first3 terms.

19 ,

13 , 1

2. Find the 1st term of a GS with g5 = 162 and r = −3.

2

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Geometric Sequencesgn = g1 · rn−1

A geometric sequence can also be expressed explicitly as:

gn = g1 · rn−1

Examples

1. If the 8th term of a GS is 243 and the 5th term is 9, write the first3 terms. 1

9 ,13 , 1

2. Find the 1st term of a GS with g5 = 162 and r = −3.

2

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Geometric Sequencesgn = g1 · rn−1

A geometric sequence can also be expressed explicitly as:

gn = g1 · rn−1

Examples

1. If the 8th term of a GS is 243 and the 5th term is 9, write the first3 terms. 1

9 ,13 , 1

2. Find the 1st term of a GS with g5 = 162 and r = −3.

2

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Geometric Sequencesgn = g1 · rn−1

A geometric sequence can also be expressed explicitly as:

gn = g1 · rn−1

Examples

1. If the 8th term of a GS is 243 and the 5th term is 9, write the first3 terms. 1

9 ,13 , 1

2. Find the 1st term of a GS with g5 = 162 and r = −3. 2

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Sum of a Geometric Sequence

The sum of n terms of any geometric sequence is given by theformula:

Sn =g1(1− rn)

1− r, r 6= 1

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Geometric SequencesExercises

1. Find the value of k so that 2k+2,5k− 11, and 7k− 13 will form ageometric sequence.

2. Insert four geometric means between 254 and 8

125 .

3. A man accepts a position at P360,000 a year with theunderstanding that he will receive a 2% increase every year. Whatwill his salary be after 10 years of service?

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Homework 7

Vance p. 311 numbers 3, 4, 9, 11, 14, 15, 17, 18, 22, 23

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