If H1, H2, H3 Hn is the Harmonic Mean between a b, Show - Maths - Sequences and Series

Question

If H1, H2, H3 Hn is the Harmonic Mean between a & b, Show that
( H1+a) / ( H1-a) + ( Hn+ b) / ( Hn - b) = 2n

Manbar Singh · Accepted Answer

Let H1, H2, H3,
,Hn be n Harmonic means between a and b
Then a, H1, H2, H3,
,Hn, b are in HP
now, 1a, 1H1, 1H2, 
, 1Hn , 1bare in AP
now, 1b = n+2th  term of AP = 1a + n+2-1d, where d is common difference of AP
now, d = 1b - 1an+1 = a-babn+1Now, 1H1 = 1a + d = 1a +  a-babn+1 = bn + aabn+1⇒H1 = abn+1bn+a     
1Now, 1H2 = 1a + 2d = 1a +  2a-babn+1 = 2a+n-1babn+1⇒H2 = abn+12a+n-1bContinuing in the same manner, we getHn = abn+1an+b       
2From 1,     H1a = bn+bbn+a⇒H1+aH1-a = 2bn+a+bb-a    using componendo and dividendo     
3From 2, Hnb = an+aan+b⇒Hn+bHn-b = 2an+a+ba-b      using componendo and dividendo     
4adding 3 and 4, we getH1+aH1-a + Hn+bHn-b = 2bn+a+bb-a + 2an+a+ba-b = 2bn+a+bb-a - 2an+a+bb-a⇒H1+aH1-a + Hn+bHn-b = 2bn+a+b-2an-a-bb-a = 2nb-ab-a = 2n

If H1, H2, H3......Hn is the Harmonic Mean between a & b, Show that ( H1+a) / ( H1-a) + ( Hn+ b) / ( Hn - b) = 2n