 Hello and welcome to the session. In this session we discuss the following question that says insert four harmonic means between 22 and 2. The first we have that is h1, h2, to hn harmonic progression that is hp then we say h1, h2 and so on up to hn are defined to be n harmonic means between a and b. Also if we have that the sequence abc and so on harmonic progression that is hp then the reciprocals of these terms that is 1 upon a, 1 upon b, 1 upon c, harmonic progression ap. Then ap is equal to a plus n minus 1 the whole into d. This is the key idea that we use for this question. Let's now move on to the solution. We are supposed to insert four harmonic means between 22 and 2. So, call this let us suppose h3 and h4 refer harmonic means between 22 so this means that 22, h1, h3, harmonic progression. Now, since this is a harmonic progression therefore the reciprocal of the tonic progression would form an ap that means 1 upon 22, 1 upon h1, 1 upon h2, 1 upon h3, 1 upon h4, 1 upon 2, rn, ap. Take 1 upon h1 as 1 upon h3 as a3, 1 upon h4 as a4. So, this means that 1 upon 22, a1, a2, a3, 1 upon 2, ap. So, this means the first term is equal to 1 upon 22 is let d be the common d and a4. You can see in this ap where total of ap is 1 upon 2 that means the first term of the ap that is 1 upon 22 the common difference which is d is equal to 1 upon 2. So, from here we have 5d is equal to 1 upon 2 minus 1 upon 22 or you can say 5d is equal to 22 as the lc of 11 minus 1 which is equal to 10 upon 22. This gives us d is equal to 10 upon 22 into 1 upon 5 times is 22. So, this is equal to 1 upon 11 d which is the common difference is equal to 1 upon 11. Now, we can easily find out the terms a1, a2, a3 and a4 of the ap. Now, this is equal to a which is 1 upon 22 minus 1 which is 1 into d which is 1 upon 11 and this is equal to this is equal to 3 upon 22. 1 which is the second term of the ap is 3 upon 22 this is equal to a which is 1 upon 22 plus this is the third term. So, here m would be 3 then which is d. So, 22 as the lcm here we have 1 plus 4 which is equal to 5 upon 22. The sum of the ap which is a2 is equal to 5 upon 20. So, minus 1 which is 3 into 1 upon 11 we are taking 22 as lcm here we have 1 plus 6 which is equal to 7 upon 22 that is the fourth term of the ap is 7 upon 24 which is the fifth term of the ap this is equal to 1 upon 22 into 1 upon 11 which is the common difference of 22 and where we have 1 plus 8 which is equal to 9 upon 22 thus we now have 9 upon 20 to 1 upon h2 as a2, a3 is for a1, a2, a3 and a4. So, we can easily find out the values for h1 h2 h3 and h4 this means that h1 is equal to reciprocal of a1 and the value of a1 is upon 22 a1 would be 22 upon 3 which means h1 is equal to 22 upon 3 in the same way h2 would be the reciprocal of the value of a2 is so h2 would be the reciprocal of 5 upon 22 which is 22 upon 5 then reciprocal of a3 h3 is equal to 22 upon h4 would be the reciprocal of upon 22 so its reciprocal would be 22 upon 9 got the values for h1 h2 h3 and h4 and assumed h1 h2 h3 h4 be the four harmonic means between 22 and 2 22 22 upon 3 22 upon 5 22 upon 7 22 upon 9 harmonic progression so we are required on 7 22 upon 10 hope you have understood the solution of this question