Question 974404
If a,b,c, are in harmonic progression shown that (1/a+1/b-1/c)(1/b+1/c -1/a)=4/ac-3/b^2
<pre>
{{{matrix(1,17,

x-d, "," , x, "," , x+d,can, represent,any, arithmetic, progression,with,common,difference,d,and, 3,terms)}}}

{{{matrix(1,14,

Therefore,1/(x-d), "," , 1/x, "," , 1/(x+d),can, represent, any,harmonic, progression,with,3,terms)}}}

{{{matrix(1,6,

Let,a=1/(x-d), "," ,b=1/x, "," ,c=1/(x+d))}}}

{{{matrix(1,6,

Therefore,1/a=x-d, "," ,1/b=x, "," ,1/c=x+d)}}}

The left side

{{{(1/a+1/b-1/c)(1/b+1/c -1/a)}}} becomes

{{{((x-d)^""+x-(x+d)^"")(x+(x+d)^""-(x-d)^"")}}}{{{""=""}}}{{{(x-d+x-x-d)(x+x+d-x+d))}}}{{{""=""}}}{{{(x-2d)(x+2d)}}}{{{""=""}}}{{{x^2-4d^2}}}

----------------------------------------------------------------

The right side:

{{{4/(ac)^""}}}{{{""-""}}}{{{3/b^2}}}{{{""=""}}}

{{{4*(1/a)^""*(1/c)^""}}}{{{""-""}}}{{{3*(1/b)^2}}} becomes

{{{4*(x-d)*(x+d)-3(x)^2}}}{{{""=""}}}{{{4(x^2-d^2)-3x^2}}}{{{""=""}}}{{{4x^2-4d^2-3x^2}}}{{{""=""}}}{{{x^2-4d^2}}}

Edwin</pre>