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