[Music] in this video we will discuss harmonic sequence so where do we usually heard the word harmonic we usually heard this word in math and in music the term harmonic is often associated with musical sounds and strange instruments a tone which is twice the frequency of the fundamental frequency is called the second harmonic while a tone which is twice the frequency of the fundamental frequency is what we call the third harmonic all right so example i have here a guitar in a guitar the length of the string is proportional to the number of vibrations of the string per second but it's actually proportional to the number of vibrations of the string per second so does a set of strings whose length are proportional produces a harmonious sound if x is the length of the string of a guitar then the sequence of the lengths of the string is x x over two x over three and x over four and so on or it produces a sequence a harmonic sequence which is one one half one third and one fourth and so on so we have the harmonic sequence one one half one third and one fourth now if we will be getting the reciprocal of this harmonic sequence we will have one two three four and so on so when we say reciprocal um a tongue denominator nothing numerator and then numerator no need to write once adding denominator all right and uh if as you can see one two three four is an example of arithmetic sequence so what do you think is the relationship of these two all right so let's define harmonic sequence uh how does a arithmetic sequence connected to harmonic sequence okay so it this is a sequence whose reciprocals form an arithmetic sequence which is called a harmonic sequence so in other words an um army uh harmonic sequence is just the reciprocal of the arithmetic sequence so a four hostile dance award no reciprocal synaptic harmonic sequence this is just the reciprocal of your arithmetic sequence all right so the n term of a harmonic sequence is given by so also we will be using the formula in finding the nth term of the liquid arithmetic sequence so in finding the n term of the arithmetic sequence we are using the formula a sub 1 plus the quantity of n minus 1 times the d or the common difference now since we are getting the harmonic sequence we will just use the reciprocal of the formula of finding the n term of the arithmetic sequence and that is 1 over a sub 1 plus the quantity of n minus 1 times d which is your common difference so as you can see this is just the reciprocal of the formula of finding the n term of your arithmetic sequence so reciprocal so this is the formula for harmonic sequence all right so let's have first an activity so let's see if you can get the a harmonic sequence given b given the arithmetic sequence so i have here 1 5 9 13 okay so for us to find the next terms of the given arithmetic sequence of course we have to look for d or common difference we have to subtract uh your first term from your second term or a sub 2 minus a sub 1 okay so 5 minus 1 that is 4 so therefore our common difference or jung di nathan is equal to 4 so nothing so second term minus your first term all right so 13 so what do you think is the next term so 13 plus 4 that is 17 okay let us apply the harmonic sequence which is the reciprocal of your arithmetic sequence so we will have one okay since the reciprocal is one is also one is still one okay and then one over five one over nine one over thirteen since our next terminating detail is seventeen so therefore our next term didn't do i one over seventeen so next we have six two negative two negative six so for us to find the next term how uh what are you going to find you have to look for the common difference which is your d so two minus six that is negative four so the d or your common difference is negative four so therefore the next term here is negative six plus negative four that is negative ten okay so the next term here is negative ten so let us get the harmonic sequence so it's just the reciprocal so one over six one over two negative one over two negative one over six now this since this is negative ten so young next in detail is negative 1 over 10. next i have 7 15 over 2 8 17 over 2 and so on so how are we going to find the common difference so again 15 over 2 minus seven so that is one half but nothing one half because say we will have a 15 over two minus seven so you can um unsha find the lcd and then subtract okay so the common difference is one half so hindi nothing solution okay so one half uncommon difference nothing so therefore 17 over two plus one half since similar fraction sila so 17 plus 1 that is 18 and then copy the common denominator which is 2. so 18 divided by 2 therefore um next not indeed the next term nothing is 9 okay so let us find the comma a harmonic sequence of the given arithmetic sequence so that is one over seven reciprocal so remember um and denominator so therefore a reciprocal is 2 over 15 and then this one since this is 8 over 1 so 1 over 8 and then this one 2 over 17 so since i'm next not in d to i so remember that your harmonic sequence is just the reciprocal of your arithmetic sequence that's why these two are related to each other okay so let's have an example find the next two terms of the harmonic sequence so you are asked to find the next two terms of the harmonic sequence so again for us to find the next terms sequence harmonic sequence is just a reciprocal of arithmetic sequence difference so 26 minus 30 that is negative 4 so therefore 22 plus negative 4 that is 18 and then 18 plus negative 4 that is 14 okay 18 that is 1 over 18 and reciprocal 14 that is 1 over 14. so find the next two terms the next two terms are one over eighteen and one over fourteen next so i have here four over fifteen two over nine four over twenty one one over six so you are asked to find the next two terms so again arithmetic sequence which is the reciprocal so we will have 15 over 4 9 over 2 21 over 4 since this is 1 over six so six nalang sha okay so nothing find the common difference so nine over two minus fifteen over four okay so it's not end so actually when you are solving fractions you can make use of the lcd or the butterfly method okay so um parameters 9 times 4 that is 36 and then this one 15 times 2 that is 30 so 36 minus 30 that is 6 and then 2 times 4 that is 8. so multiply neutral multiplying multiplied so this is 36 minus 30 that is 6 over 2 times 4 that is 8. so 6 over 8 that is three-fourths pagner just nothing six over eight that is three-fourths so therefore and the common difference nito i three fourths omega attack and three fourths are six so six plus three-fourths six times four that is twenty-four and then three times one divide my pattern one taijan so three times one so 24 plus 3 times 1 3 so 24 plus 3 that is 27 and then 1 times 4 that is 4 so 27 over 4. so this is 27 over 4. next italian and 20's and 3 27 over four now since similar fraction the numerator that is 27 plus 30 at plus 3 that is 30 and then copy the common denominator which is 4 so 30 over 4 but the pinch reduce the lowest term so that is 15 over 2. so therefore this is 15 over 2. now this is the arithmetic sequence 27 and 2 over 15. let's have another example so find the eighth term of the harmonic sequence so i have here one-half one-fourth one over six and so on so first get the arithmetic the reciprocal or the arithmetic sequence of the given or harmonic sequence and that is 2 4 6. so again when we are getting the arithmetic sequence we are just getting its reciprocal okay so let us identify a sub 1 since we are getting the eighth term so masha dershong malayo we can apply the formula uh the formula for the arithmetic finding the n term of arithmetic sequence so since harmonic sequence it or reciprocal so a sub 1 over a sub 1 is 2 our d the common difference is 2 has a 4 minus 2 that is 2 and then our n since 8 terms so our n is 8. now let us apply the formula of finding the nth term of our harmonic sequence so again this is the formula in finding the n term for arithmetic sequence now since harmonic okay so let us substitute the values so we will have 1 over your a sub 1 is 2 plus your n is 8 and then minus one your common difference is two so eight minus one that is seven seven times two is fourteen fourteen plus two so we will have one over sixteen so therefore the eighth term is one over sixteen so eighth term nothing is one over sixteen next find the twenty-fifth term of this harmonic sequence so i have one fourth one over fourteen one over twenty four and so on so we will have so nate reciprocal that is four fourteen twenty four so we are looking for the twenty-fifth term so let us identify first a sub one which is your fourth so your first term d the common difference is 10 y 10 because 14 minus 4 that is 10 and then your n is 25 since we are looking for the 25th term okay um by uh you're applying the formula so we have 1 over a sub 1 plus the quantity of n minus 1 times the common difference so substitute the values we have one your a sub 1 is 4 your n is 25 minus 1 your common difference is 10. so 25 minus 1 that is 24 times 10 that is 240 plus 4 so we will have 1 over 244 so therefore our 25th term is 1 over 244 hope you learned something don't forget to like subscribe and hit the notification bell for updated ko for more video tutorials this is your guide in learning your math lessons your walmart channel