If H1,H2,…,Hn are n harmonic means between a and b(≠a), then value of H1+aH1−a+Hn+bHn−bis equal to

As a,H1,H2,…,Hnb are in H.P.    1a,1H1,1H2⋯1Hn,1b are in A.P.Let dbe the common difference of this A.P., then            1b=1a+(n+1)dand  1Hn−1H1=(n−1)dNow, H1+aH1−a=1/a+1/H11/a−1/H1=1/a+1/H1−dand Hn+bHn−b=1/b+1/Hn1/b−1/Hn=1/b+1/Hnd∴    H1+aH1−a+Hn+bHn−b=1/a+1/H1−d+1/b+1/Hnd                                  =1d1b−1a+1Hn−1H1=2n