If `a^2(b+c),b^2(c+a),c^2(a+b),` are in A.P. show that either `a ,b ,c` are in A.P., or `a b+b c+c a=0.`

Explanation

` a^(2) (b + c), b^(2) (c + a) , c^(2) (a + b) ` , are in A.P.
` rArr b^(2) (c +a) - a^(2) (b + c) = c^(2) (a + b) - b^(2) (c + a)`
` rArr (b^(2) a - a^(2) b) + (b^(2) c - a^(2) c) = (c^(2) b - b^(2) c) + (c^(2) a - b^(2) a)`
` rArr (b -a) (ab + bc + ca) = (c-b) (ab + bc + ca)`
`rArr (2b - a - c) (ab + bc + ca) = 0 `
Hence a, b, c are in A.P. or ` ab + bc + ca = 0 `