Harmonic Sequence (Tagalog/Filipino Math)

Summary

This video tutorial introduces harmonic sequences, which are sequences of reciprocals of an arithmetic sequence. It explains the definition, provides examples, and demonstrates how to find specific terms and the harmonic mean. The tutorial covers how to calculate the nth term of a harmonic sequence and how to insert harmonic means between given numbers. It also solves problems involving finding the first term of a sequence and the common difference.

Takeaways

• 😀 A harmonic sequence is formed by taking the reciprocals of the terms of an arithmetic sequence.
• 🎓 The general formula for the nth term of a harmonic sequence, derived from an arithmetic sequence with first term a and common difference d, is 1 / (a + (n-1)d).
• 📐 Examples given in the script illustrate how to find specific terms in a harmonic sequence by applying arithmetic sequence properties to their reciprocals.
• 🔍 The script demonstrates how to calculate the harmonic mean of two or more numbers using the formula H = 2ab / (a+b) for two numbers, and extending it for more numbers.
• 📘 The tutorial explains the process of finding the first term of a harmonic sequence when given other terms, by solving a system of equations derived from the arithmetic sequence of their reciprocals.
• 📊 The script provides a method to insert a specific number of harmonic means between two given numbers by determining the common difference of the underlying arithmetic sequence.
• 🧮 Practical examples are used to show how to calculate the harmonic mean of numbers like 24 and 12, and a series like 3, 4, 5, using the harmonic mean formula.
• 📖 The tutorial covers how to find a term in a harmonic sequence by setting up and solving equations based on the properties of the corresponding arithmetic sequence of the reciprocals.
• 📐 The script also explains how to determine the number of terms in a harmonic sequence by using the formula for the nth term of an arithmetic sequence and solving for n.
• 🎯 The tutorial concludes with a summary of the key concepts and formulas related to harmonic sequences, reinforcing the understanding of how to analyze and work with them.

Q & A

What is a harmonic sequence?

A harmonic sequence is a sequence formed by taking the reciprocal of each term in an arithmetic sequence.

How is the harmonic sequence related to the arithmetic sequence?

The harmonic sequence is related to the arithmetic sequence by taking the reciprocal of each term in the arithmetic sequence.

Can you provide an example of a harmonic sequence?

An example of a harmonic sequence is 1, 1/2, 1/3, 1/4, which are the reciprocals of the arithmetic sequence 1, 2, 3, 4.

What is the formula to find the nth term of a harmonic sequence?

The nth term of a harmonic sequence can be found using the formula 1/(a + (n-1)d), where 'a' is the first term and 'd' is the common difference of the corresponding arithmetic sequence.

How do you find the harmonic mean of two numbers?

The harmonic mean of two numbers 'a' and 'b' is calculated using the formula 2ab / (a + b).

What is the relationship between the harmonic mean and the arithmetic mean?

The harmonic mean is always less than or equal to the arithmetic mean for any set of numbers.

Can you give an example of how to find the 15th term of a harmonic sequence?

To find the 15th term of the harmonic sequence 2, 6/11, 11/15, 19/28, ..., you first determine the corresponding arithmetic sequence and then use the formula for the nth term of an arithmetic sequence.

How many harmonic means can be inserted between 1/2 and 1/52?

Four harmonic means can be inserted between 1/2 and 1/52, resulting in the sequence 1/2, 1/12, 1/22, 1/32, 1/52.

What is the first term of a harmonic sequence if the third term is 1/13 and the twentieth term is 1/64?

The first term of the harmonic sequence can be found by setting up a system of equations using the formula for the nth term of an arithmetic sequence and solving for the first term.

How do you determine if a sequence is a harmonic sequence?

A sequence is a harmonic sequence if its terms are the reciprocals of an arithmetic sequence, which can be verified by checking if the differences between the reciprocals of consecutive terms are constant.

Outlines

00:00

📚 Introduction to Harmonic Sequences

The video begins with an introduction to harmonic sequences, a type of sequence derived from arithmetic sequences. The host explains that if you have an arithmetic sequence, the sequence of the reciprocals of its terms forms a harmonic sequence. Several examples are given to illustrate this concept, such as the sequence 1/2, 1/3, 1/4, which is the harmonic sequence derived from the arithmetic sequence 2, 3, 4. The video then proceeds to demonstrate how to identify and prove a sequence as harmonic by examining its reciprocals and common differences.

05:00

🔍 Analyzing Harmonic Sequences with Examples

This section delves into analyzing harmonic sequences through various examples. The host calculates the seventh term of a harmonic sequence and explains the process of finding terms in a harmonic sequence by using the arithmetic sequence properties of their reciprocals. The video also covers how to find a specific term in a harmonic sequence by setting up equations based on the properties of arithmetic sequences of their reciprocals. The examples include finding the 15th term of a given harmonic sequence and determining the term that equals 1/345 in another sequence.

10:01

🧮 Solving for Terms and Means in Harmonic Sequences

The video continues with solving for the first term of a harmonic sequence given the third and twentieth terms. The host uses a system of equations to find the unknown first term by transforming the harmonic sequence into its reciprocal arithmetic sequence. The concept of harmonic mean is introduced, and the formula for calculating the harmonic mean of two and three numbers is explained with examples. The video demonstrates how to calculate the harmonic mean of 24 and 12, and then of the numbers 3, 4, and 5.

15:11

📈 Inserting Harmonic Means in a Sequence

The final part of the video script discusses the process of inserting harmonic means between two given terms of a harmonic sequence. The host calculates the common difference of the sequence by using the terms one half and one over fifty-two. Four additional harmonic means are then calculated to be inserted between these two terms, resulting in the sequence terms 1/12, 1/22, 1/32, and 1/42. The video concludes with a brief summary of the content covered and a sign-off, indicating the end of the tutorial on harmonic sequences.

Keywords

💡Harmonic Sequence

A harmonic sequence is a series where each term is the reciprocal of the corresponding term in an arithmetic sequence. This concept is central to the video's theme, which is to teach viewers about different types of sequences. In the script, the creator explains that if you have an arithmetic sequence, taking the reciprocal of each term results in a harmonic sequence. For example, if the arithmetic sequence is 1, 1/2, 1/3, 1/4, the harmonic sequence would be the reciprocals: 1, 2, 3, 4.

💡Arithmetic Sequence

An arithmetic sequence is a sequence of numbers with a constant difference between consecutive terms. It is mentioned in the script as a basis for creating a harmonic sequence. The video uses arithmetic sequences like 1, 2, 3, 4 to demonstrate how their reciprocals form a harmonic sequence, emphasizing the relationship between arithmetic and harmonic sequences.

💡Geometric Sequence

A geometric sequence is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. While the main focus of the video is on harmonic sequences, geometric sequences are mentioned in the context of contrasting different types of sequences, highlighting the diversity in mathematical sequences.

💡Reciprocal

The reciprocal of a number is 1 divided by that number. In the context of the video, reciprocals are used to transform an arithmetic sequence into a harmonic sequence. The script gives examples such as taking the reciprocal of each term in the sequence 1, 1/2, 1/3, 1/4 to get the harmonic sequence 1, 2, 3, 4.

💡Common Difference

The common difference in an arithmetic sequence is the constant difference between consecutive terms. The script explains how to find the common difference in a harmonic sequence by looking at the differences between the reciprocals of the terms in the corresponding arithmetic sequence.

💡Arithmetic Mean

The arithmetic mean is the average of a set of numbers. It is briefly mentioned in the script when comparing it to the harmonic mean, which is the focus of the video. The arithmetic mean is calculated by summing all the numbers and dividing by the count, whereas the harmonic mean is calculated differently and is used in specific contexts like physics and engineering.

💡Harmonic Mean

The harmonic mean is a type of average that is particularly useful for sets of numbers that represent rates or ratios. The video provides a formula for calculating the harmonic mean and uses it to solve problems. For example, the harmonic mean of 24 and 12 is calculated as 16, which is a key concept in the video.

💡Sequence

A sequence is an ordered list of objects or numbers. The video is entirely focused on sequences, specifically harmonic sequences, but also touches on arithmetic and geometric sequences. Sequences are fundamental to mathematics and are used in various fields, including physics, engineering, and computer science.

💡Formula

A formula in mathematics is a concise way of expressing information symbolically. The video script includes formulas for calculating terms in a sequence and for finding the harmonic mean. Formulas are essential tools for solving mathematical problems and are used throughout the video to illustrate how to work with harmonic sequences.

💡Term

In the context of sequences, a term refers to an individual element within the sequence. The script discusses finding specific terms in a harmonic sequence, such as the seventh term or the fifteenth term, using arithmetic sequence properties. Understanding terms is crucial for grasping how sequences are constructed and manipulated.

💡Tutorial

A tutorial is a set of instructions or an explanation that helps people understand how to do something. The video is described as a tutorial, meaning it is designed to educate viewers on the concept of harmonic sequences. The script is structured to teach by providing definitions, examples, and step-by-step problem-solving.

Highlights

Introduction to harmonic sequences as the sequence of reciprocals of an arithmetic sequence.

Definition of a harmonic sequence and its relation to arithmetic sequences.

Illustration of a harmonic sequence with the example 1, 1/2, 1/3, 1/4, and explanation of its properties.

Explanation of how to prove a sequence is harmonic by demonstrating the common difference in the reciprocals.

Tutorial on finding the seventh term of a harmonic sequence using the arithmetic sequence formula.

Example calculation of the 15th term of a harmonic sequence and the method used.

Methodology for transforming a given arithmetic sequence into a harmonic sequence and finding a specific term.

Process of finding the first term of a harmonic sequence given the third and twentieth terms.

Calculation of the harmonic mean of two numbers, 24 and 12, using the harmonic mean formula.

Tutorial on calculating the harmonic mean of three numbers: 3, 4, and 5.

Guide on how to insert four harmonic means between 1/2 and 1/52.

Explanation of the common difference in a harmonic sequence and how it's used to find intermediate terms.

Final thoughts and summary of the tutorial on harmonic sequences.

Transcripts

hi guys welcome to engineered my channel

harmonic sequence

just keep on watching

[Music]

okay so this time a tutorial another

type of

sequence which is the harmonic sequence

submarine upon previous videos about

different sequence like arithmetic

sequence and geometric sequence

as well as young corresponding series

arithmetic art geometric series

in definition so harmonic sequence so

sub

if a sub 1 comma a sub 2 comma a sub 3

comma dot dot dot are terms in an

arithmetic sequence

then the sequence of reciprocal of these

terms 1 over e sub 1

comma 1 over e sub 2 comma 1 over e sub

3 comma delta dot comma until 1 over e

sub n

is called the harmonic sequence okay so

hyacinth

related young harmonic sequence

arithmetic sequence

arithmetic sequence of one e sub three e

sub two

and so on

unintended harmonic sequence

harmonic sequence is arithmetic sequence

para must have been on the harmonic

sequencing

given sequence okay so to illustrate

harmonic sequence

examples one comma one half comma one

third comma one fourth

so by definition

harmonic sequence

arithmetic sequencing reciprocal right

with a common ratio of

tagging one so therefore you see

procalnito which is this harmonic

sequence is proven

in a harmonic sequence next

one over fifteen one over eleven one

over seven one over three

suppose we have fifteen

eleven seven and three so nothing

common difference so i know negative

four right

harmonic sequence so therefore proven

the harmonic sequential

okay next two comma one comma two thirds

comma one half and so on so reciprocal

nothing

one half one three halves

two so my common difference basila

so one minus one half is one half three

halves minus one

is one half two minus three halves is

one nap

so therefore you're gonna see procalnito

is made uncommon

differential arithmetic sequence so

therefore this is an example also of

harmonic sequence

okay so parameters

analyzing any harmonic sequence is

mixol10

examples for the first one we have find

the seventh term

of the harmonic sequence one half comma

one

seven comma one over twelve comma so on

okay

is two young one over seven is seven

one over twelve is twelve right so the

timeline

lies number seven term atma probably not

mythic sequencing reciprocal right

parameters having harmonic

sounding common difference in la 7 minus

2 is 5

12 minus 7 is five summation common

difference or d in a five

okay tapasya first term is two so using

the formula for the n term of arithmetic

sequencing without

a sub n is equal to e sub 1 plus

n minus 1 times d right so pacman log in

add a new value e sub 1 is 2

plus n minus 1 times d 5

okay so that is nothing behind a ping

seventh term

hana pentane seventh term nitong

arithmetic sequence so a sub seven is

equal to two plus

seven so seven minus one times five

so we have two plus seven minus one is

six times five

or two plus thirty right or thirty two

so therefore young's seventh term niton

arithmetic sequence nathan is thirty two

so therefore the seventh term of the

harmonic sequence is

one over thirty two okay next we have

find the 15 term of the harmonic

sequence

2 and 6 over 11 comma 113 over 15 comma

1 and 9 over 19 comma that that that

okay so this time given diagonal mixed

number

11 is what eleven times two twenty two

plus six twenty it's a twenty eight over

eleven

and then one thirty number fifteen

fifteen times one is fifteen plus

thirteen

is twenty eight over fifteen

ten 19 times 1 is 19 plus 9 is 28

over 19. so reciprocal

in bali we have 11 over 28 15 over

28 19 over

15 over 28 minus 11 over 28

4 over 28 19 over 28 minus 15 over 28

4 over 28

n is equal to a sub 1 plus n minus 1

times d

so e sub one give it anything else

arithmetic sequence eleven over twenty

eight

plus n minus one times d

uncommon difference now four over twenty

eight nine

15 minus 1 times 4 over 20

so 11 over 28 plus 15 minus 1 is 14

so 14 times 4 is what

56 56 over 28 so add nothing

similar fraction so not in your

numerator so 11 plus 56

is 67 and then copy the same denominator

arithmetic sequence

harmonics so

28 over 67

okay so therefore the 15 terminal adding

harmonic sequence is 28 over

67 okay

next we have in the harmonic sequence

one half comma one over nine comma one

over sixteen comma one over twenty three

comma delta dot

which term is one over three four five

okay so nugget transformed an atom in a

given

terms new harmonic sequence into

reciprocal so we have 2

9 16 23

comma dot dot so therefore

reciprocal is arithmetic sequence

e sub n is equal to e sub 1 which is 2

plus n minus 1 times unknown common

difference 10 minus 2

is 7 okay so

i'm given though not an issue one over

three

four five so long term so

nothing reciprocal in three four

five right so given value

and term which is three four

four five minus two equals n minus one

times seven

three four five minus two is three four

three equals

n minus one times seven so divide both

sides by seven

elemental

equal to 340 divided by 7 is

49 then cancel d to c 7 right so my

gigging 49 is equal to n

minus 1 then transpose cone element is

negative 1

14 n plus 1 equals n or n therefore is

equal to forty nine plus one or

fifty so therefore fifty-eighth term

one over three four five nineteen

harmonic sequence

okay next we have the third term of a

harmonic sequence

is one over thirteen and the twentieth

term is one over sixty-four

find the first term of that sequence

okay so in given thousand third term or

e sub three no harmonic sequence now one

over

thirteen at you a sub 20

now one over sixty four so like a

transformation into

reciprocals thirteen right

anita gigging

equals 13 and e sub 20 equals 64.

so the by is equal to a sub 1 plus

n minus 1 times t unknown

two systems of equations into unknown

so not indeed to say 13 say sub n

equal e sub one unknown plus n value is

three case e sub three so three minus

one times

b is also unknown so we have 13 is equal

to e sub one plus

three minus one is 2d equation

1 then determines a sub 20 so 64 is

equal to e sub 1 plus

n value is 20 so 20 minus 1 times d

or 64 is equal to e sub 1 plus 20 minus

1 is 19 times d

equation 2 so per nothing is subtract so

subtract equation 2 the 64 equals c sub

1 plus 19 d

minus equation 1 the 13 is equal to e

sub 1

plus 2 d so 64 minus 13 is what

51.

is

b so dividing both sides by 17

i know must have solved nothing but

unity 51 divided by 17

is three right so therefore alumni

but by using

any of these two equations so you get it

in the round 13 is equal to e sub 1

plus 2 times 3

13 is equal to e sub 1 plus 2 times 3

over 6

transposes six thirteen minus six is

equal to e sub one

so therefore e sub one is thirteen minus

six or

seven so sub one so corresponding

arithmetic sequence

terminal

so therefore the first term of the

harmonic sequence is one over seven

okay next we have find the harmonic mean

of twenty four

and twelve okay so just like the

arithmetic mean chaka geometric mean

burnt entire formula for harmonic mean

so the formula for harmonic mean is

let's say harmonic mean is equal to

2 a b over a plus

b

harmonic mean so this time we have let's

say a is 24 and b

is 12. so plug in length as a formula

2 times 24 times 12

divided by 24 plus 12

okay so using calculator

and nothing harmonic mean is

16 okay

next we have find the harmonic mean of

three

four five okay so this time pinappahan

optimum satin is harmonic meaning that

long numbers

so mandinthal formuladito so harmonic

mean for three numbers is equal to

three a b c over

a b plus a c plus b c

all right so let's say a c three b c

four

at c c five so plug in long nothing due

to formula so 3 times

3 times 4 times 5

over a b so 3 times 4

plus ac so 3 times 5

plus bc so four times

five okay so you still have a calculator

parameter harmonic mean

using calculator the harmonic mini squat

180 over 47

okay next we have insert four harmonic

means

between one half and one over

fifty two okay so let's say first term

nothing on one half

2 and 52 right so therefore

illinois in terms of one two three four

five six so six so therefore man

a sub six now fifty two at

a sub one na

sub 1 plus n minus 1 times d

so plug in at an a sub 6 of 52

then a sub 1 is 2 plus and nothing is 6

right so 6 minus 1 times d

so sub may not in cd so 52 is equal to 2

plus 6 minus 1 is 5 times d

transpose square d to c2 52 minus 2 is

equal to 5 d

so 52 minus 2 is 50 equals

5 d divide both sides by 5

therefore d is equal to 10 okay so

nothing in the common difference depends

is 1 over 12 1 over 22

1 over 32 and 1 over 42

okay so therefore the four harmonic

means between one half and one over

two are one over twelve one over twenty

two one over thirty two and one over

forty two okay

okay so i think that's it for this video

harmonic sequence so model number

m

[Music]

[Music]

you