If a+b,b+c,c+a are in HP then prove that 1/a^2,1/b^2,1/c^2 are also in hp

a^2. , b^2. , c^2. are in A.P.

by adding ab+b.c+c.a. in. each. term.

a^2+ab+b.c+ca. , b^2+a.b+b.c+c.a. , c^2+a.b+b.c+c.a. are in A.P.

(a+b)(c+a). , (a+b) (b+c). , (b+c)(c+a). are in A.P.

On dividing by (a+b)(b+c)(c+a) in each term

or. 1/(b+c). , 1/(c+a). , 1/(a+b). are in A.P. , proved.