Harmonic Sequence (Tagalog/Filipino Math)
Summary
TLDRThis 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
📚 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.
🔍 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.
🧮 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.
📈 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.
Mindmap
Keywords
💡Harmonic Sequence
💡Arithmetic Sequence
💡Geometric Sequence
💡Reciprocal
💡Common Difference
💡Arithmetic Mean
💡Harmonic Mean
💡Sequence
💡Formula
💡Term
💡Tutorial
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
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