Question

If a b c are in A.P then show that ab(a+b),bc(b+c),ca(c+a) are also in A.P


Please help me it's urgent​

Answer 1

Answer:

∵a,b,careinAP

hence,(b−a)=(c−b)⟶(1)

(i) ∵b

2

(c+a)−a

2

(b+c)

=(b−a)(ab+bc+ca)

andc

2

(a+b)−b

2

(c+a)

=(c−b)(ab+bc+ca)

=(b−a)(ab+bc+ca)(from(1))

hence,b

2

(c+a)−a

2

(b+c)=c

2

(a+b)−b

2

(c+a)

∴a

2

(b+c),b

2

(c+a),c

2

(a+b)arealsoinAP

(ii) ∵(c+a−b)−(b+c−a)

=2(a−b)

=−2(b−a)

and(a+b−c)−(c+a−b)

=2(b−c)

=−2(c−b)

=−2(b−a)(from(1))

hence(c+a−b)−(b+c−a)=(a+b−c)−(c+a−b)

∴(b+c−a),(c+a−b),(a+b−c)areinAP

(iii) ∵(ca−b

2

)−(bc−a

2

)

=(a−b)(a+b+c)

=−(b−a)(a+b+c)

and(ab−c

2

)−(ca−b

2

)

=(b−c)(a+b+c)

=−(c−b)(a+b+c)

=−(b−a)(a+b+c)(from(1))

hence,(ca−b

2

)−(bc−a

2

)=(ab−c

2

)−(ca−b

2

)

∴(bc−a

2

),(ca−b

2

),(ab−c

2

)areinAP