Question

If a²(b + c)
, b²(c + a), c²(a + b) are in A.P., show that either a, b, c are in A.P. or ab+be+ca=0.​

Answer 1

Answer:

x,y,z are in A.P. , xy=z−y

a

2

(b+c)−a

2

(b+c)=c

2

(a+b)−b

2

(c+a)

a

2

b+a

2

c−a

2

b−a

2

c=c

2

a+c

2

b−b

2

c−b

2

a

(a

2

b−a

2

c)+(b

2

a−a

2

b)=(c

2

a−b

2

a)+(c

2

b−b

2

c)

c(b

2

−a

2

)+ab(b−a)=a(c

2

−b

2

)+bc(c−b)

(b−a)[c(b+a)+ab]=(c−b)[a(c+b)+bc]

(b−a)(bc+ac+ab)=(c−b)(ac+bc+ab)

Either ab+bc+ac=0,

b−a=c−b