`text(Definition :)`
If `a, b, c` are in `H.P.` then middle term is called the harmonic mean between them. Hence if `H` is the harmonic mean (`H .M.`) between `a` and `b` then `a, H, b` are in `H.P.` and `H =(2ab)/(a+b)`.
(Recall that `AM = (a+b)/2` and `GM = sqrt(ab)` if `a > 0, b > 0`)
`text(To insert 'n' HM's between a and b :)`
Let `H_1, H_2 ........ H_n` are `n` `HM's` between `a` and `b`
hence `a, H_1, H_2, ..... H_n, b` are in `H.P.`
`1/a,1/H_1,1/H_2...................... 1/H_n,1/b` are in `A.P.`
`1/b=1/a+(n+1)d; -1/a=(n+1)d; d=(a-b)/(ab(n+1))`
`1/H_1=1/a+d`
`1/H_2=1/a+2d`
`1/H_3=1/a+3d`
`vdots`
`1/H_n=1/a+nd`
`sum_(i=1)^n1/H_i=n/a+(d(n)(n+1))/2=n/a s+(n(n+1))/2 . ((a-b))/(ab(n+1))`
`=n[1/a+(a-b)/(2ab)]=n/(2ab)[2b+a-b]=(n(a+b))/(2ab)=n . 1/H`
Hence sum of the reciprocals of all then `HM's` between `a` and `b` is equal to `n` times the reciprocal of single `HM` between `a` and `b`.
`text(Definition :)`
If `a, b, c` are in `H.P.` then middle term is called the harmonic mean between them. Hence if `H` is the harmonic mean (`H .M.`) between `a` and `b` then `a, H, b` are in `H.P.` and `H =(2ab)/(a+b)`.
(Recall that `AM = (a+b)/2` and `GM = sqrt(ab)` if `a > 0, b > 0`)
`text(To insert 'n' HM's between a and b :)`
Let `H_1, H_2 ........ H_n` are `n` `HM's` between `a` and `b`
hence `a, H_1, H_2, ..... H_n, b` are in `H.P.`
`1/a,1/H_1,1/H_2...................... 1/H_n,1/b` are in `A.P.`
`1/b=1/a+(n+1)d; -1/a=(n+1)d; d=(a-b)/(ab(n+1))`
`1/H_1=1/a+d`
`1/H_2=1/a+2d`
`1/H_3=1/a+3d`
`vdots`
`1/H_n=1/a+nd`
`sum_(i=1)^n1/H_i=n/a+(d(n)(n+1))/2=n/a s+(n(n+1))/2 . ((a-b))/(ab(n+1))`
`=n[1/a+(a-b)/(2ab)]=n/(2ab)[2b+a-b]=(n(a+b))/(2ab)=n . 1/H`
Hence sum of the reciprocals of all then `HM's` between `a` and `b` is equal to `n` times the reciprocal of single `HM` between `a` and `b`.