Question

If p^th, q^th, r^th terms of a H.P. be a, b, c respectively, prove that (q – r)bc + (r – p) ac + (p – q) ab = 0 

Answer 1

Let x be the first term and d be the common difference of the corresponding A.P.

(i) - (ii) ⇒ ab(p – q)d = b – a ..........(iv) 

(ii) - (iii) ⇒ bc (q – r)d = c – b ..........(v) 

(iii) - (i) ⇒ ac (r – p) d = a – c ..........(vi) 

(iv) + (v) + (vi) gives 

bc (q – r) + ac(r – p) + ab (p – q) = 0