Question 1188899
if {{{a/b=b/c=c/d}}}, then prove that {{{(a^3+b^3+c^3)/(b^3+c^3+d^3)}}}{{{""=""}}}{{{a/d}}}
<pre>
Let {{{k=a/b=b/c=c/d}}}, then

{{{matrix(1,5,a=kb, ",", b=kc,",", c=kd)}}}

{{{a=k(kc)=k^2c=k^2(kd)=k^3d}}}

{{{b=kc=k(kd)=k^2d}}}

So, since {{{a=k^3d}}}, {{{b=k^2d}}}, and {{{c=kd}}},

{{{(a^3+b^3+c^3)/(b^3+c^3+d^3)}}}{{{""=""}}}{{{((k^3d)^3+(k^2d)^3+(kd^"")^3)/((k^2d)^3+(kd^"")^3+(d^"")^3)}}}{{{""=""}}}

{{{(k^9d^3+k^6d^3+k^3d^3)/(k^6d^3+k^3d^3+d^3)}}}{{{""=""}}}{{{(k^3d^3(k^6+k^3+1))/(d^3(k^6+k^3+1))}}}{{{""=""}}}
{{{(k^3cross(d^3)(cross(k^6+k^3+1)))/(cross(d^3)(cross(k^6+k^3+1)))}}}{{{""=""}}}{{{k^3}}}

Since {{{a=k^3d}}}, {{{k^3=a/d}}}, so

{{{(a^3+b^3+c^3)/(b^3+c^3+d^3)}}}{{{""=""}}}{{{a/d}}}

Edwin</pre>