a,b,c成等差数列,那么证明a^2(b+c),b^2(a+c),c^2(b+c)成等差数列1

Question

a,b,c成等差数列,那么证明a^2(b+c),b^2(a+c),c^2(b+c)成等差数列
1

Answer 1

a、b、c成等差数列,
∴a+c=2b,
∴a^2(b+c)+c^2(a+b)-2b^2(c+a)
=a^2c+c^2a+ab(a-2b)+bc(c-2b)
=a^2c+c^2a-2abc
=ac(a+c-2b)
=0,
∴a^2(b+c)+c^2(a+b)=2b^2(c+a),
∴a^2(b+c)、b^2(c+a)、c^2(a+b)成等差数列.