Question

已知三个数a，b，c成等差数列，那么a^2(b+c),b^2(c+a),c^2(a+b)是否构成等差数列?

Answer 1

a^2(b+c)+c^2(a+b)
=a^2b+a^2c+ac^2+c^2b
=ac(a+c)+(a^2+c^2)b
=ac*2b+(a^2+c^2)b
=b(a+c)^2
=4b^2
b^2(c+a)
=b^2*2b
=2b^3
故a^2(b+c),b^2(c+a),c^2(a+b)构成等差数列

Answer 2

a^2(b+c)+c^2(a+b)
=ac（a+c）+bc^2+a^2b
=2abc+bc^2+a^2b
=ab（a+c）+bc（a+c）
=（a+c）^2*b
=4b^3
b^2(c+a)*2=4b^3
a^2(b+c),b^2(c+a),c^2(a+b)构成等差数列