If p^th , q^th , and r^th terms of a H.P are a,b,c respectively, then prove that #(q-r) / (a) + (r-p) / (b) + (p-q)/(c) = 0?

1 Answer
Dec 17, 2017

Please see below.

Explanation:

As p^(th),q^(th) and r^(th) terms of a H.P are a,b,c

p^(th),q^(th) and r^(th) terms are 1/a,1/b,1/c of corresponding A.P.

Let the first term of this A.P. be f and common difference be d

then 1/a=f+(p-1)d ............(1)

1/b=f+(q-1)d ............(2)

and 1/c=f+(r-1)d ............(3)

Subtracting (2) from (1); (3) from (2) and (1) from (3), we ger

1/a-1/b=d(p-q) or p-q=(b-a)/(abd) ............(A)

1/b-1/c=d(q-r) or q-r=(c-b)/(bcd) ............(B)

and 1/c-1/a=d(r-p) or r-p=(a-c)/(acd) ............(C)

hence (q-r)/a+(r-p)/b+(p-q)/c

= (c-b)/(abcd)+(a-c)/(abcd)+(b-a)/(abcd)

= (c-b+a-c+b-a)/(abcd)

= 0