初中代数难题设a+b+c=0,证明:[(a^2+b^2+c^2)/2]×[(a^5+b^5+c^5)/5]=(a^7+b^7+c^7)/7.
2020-04-10
初中代数难题
设a+b+c=0,证明:
[(a^2+b^2+c^2)/2]×[(a^5+b^5+c^5)/5]=(a^7+b^7+c^7)/7.
优质解答
设u=a+b,v=ab,
则a^2+b^2+c^2=2u^2-2v
a^5+b^5+c^5=5v(u^3-uv)
a^7+b^7+c^7=-7v(u^5-2u^3v+uv^2)
∴左边=(u^2-v)[v(u^3-uv)]
=v(u^5-2u^3v+uv^2)
=右边,
故原恒等式成立.
设u=a+b,v=ab,
则a^2+b^2+c^2=2u^2-2v
a^5+b^5+c^5=5v(u^3-uv)
a^7+b^7+c^7=-7v(u^5-2u^3v+uv^2)
∴左边=(u^2-v)[v(u^3-uv)]
=v(u^5-2u^3v+uv^2)
=右边,
故原恒等式成立.