document.write( "Question 856871: If 1/(y+ax),1/(2y),1/(y+az) are in arithmetic progression.prove that k^2x,y, z are in geometric progression \n" ); document.write( "
Algebra.Com's Answer #519671 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "This is true if and only if it is also true that a = k².  You MUST\r\n" );
document.write( "be given that, even though you didn't state it.  If \"a\" is not equal\r\n" );
document.write( "to k², then your proposition is not true.\r\n" );
document.write( "\r\n" );
document.write( "So we will assume that a=k²\r\n" );
document.write( "\r\n" );
document.write( "\"1%2F%28y%2Bax%29\",\"1%2F%282y%29\",\"1%2F%28y%2Baz%29\" are in arithmetic progression.\r\n" );
document.write( "\r\n" );
document.write( "Then (2nd term)-(1st term) = (3rd term)-(2nd term)\r\n" );
document.write( "\r\n" );
document.write( "\"1%2F%282y%29-1%2F%28y%2Bax%29\"\"%22%22=%22%22\"\"1%2F%28y%2Baz%29-1%2F%282y%29\"\r\n" );
document.write( "\r\n" );
document.write( "Add \"1%2F%282y%29\" to both sides\r\n" );
document.write( "\r\n" );
document.write( "\"2%2F%282y%29-1%2F%28y%2Bax%29\"\"%22%22=%22%22\"\"1%2F%28y%2Baz%29\"\r\n" );
document.write( "\r\n" );
document.write( "\"1%2Fy-1%2F%28y%2Bax%29\"\"%22%22=%22%22\"\"1%2F%28y%2Baz%29\"\r\n" );
document.write( "\r\n" );
document.write( "Get LCD = y(y+ax)(y+az)\r\n" );
document.write( "\r\n" );
document.write( "(y+ax)(y+az) - y(y+az) = y(y+ax)\r\n" );
document.write( "\r\n" );
document.write( "Factor out (y+az) on the left side:\r\n" );
document.write( "\r\n" );
document.write( "      (y+az)[(y+ax)-y] = y(y+ax)\r\n" );
document.write( "\r\n" );
document.write( "        (y+az)(y+ax-y) = y²+axy\r\n" );
document.write( "\r\n" );
document.write( "            (y+az)(ax) = y²+axy\r\n" );
document.write( "\r\n" );
document.write( "               axy+axz = y²+axy\r\n" );
document.write( "\r\n" );
document.write( "                   axz = y² \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "To show that k^2x, y, z are in geometric progression,\r\n" );
document.write( "\r\n" );
document.write( "we must show that this is true:\r\n" );
document.write( "\r\n" );
document.write( "\"second_term%2F%28first_term%29=%28third_term%29%2F%28second%2Bterm%29\"\r\n" );
document.write( "\r\n" );
document.write( "So we have to show that \"y%2F%28k%5E2x%29\"\"%22%22=%22%22\"\"z%2Fy\"\r\n" );
document.write( "\r\n" );
document.write( "We have shown above that\r\n" );
document.write( "\r\n" );
document.write( "                   axz = y²\r\n" );
document.write( "\r\n" );
document.write( "Since we assume that a = k², we substitute k² for a\r\n" );
document.write( "\r\n" );
document.write( "                  k²xz = y²\r\n" );
document.write( "\r\n" );
document.write( "We divide both sides by k²xy\r\n" );
document.write( "\r\n" );
document.write( "                 \"%28k%5E2xz%29%2F%28k%5E2xy%29\"\"%22%22=%22%22\"\"y%5E2%2F%28k%5E2xy%29\"\r\n" );
document.write( "\r\n" );
document.write( "                 \"%28cross%28k%5E2x%29z%29%2F%28cross%28k%5E2x%29y%29\"\"%22%22=%22%22\"\"y%5Ecross%282%29%2F%28k%5E2x%28cross%28y%29%29%29\"                 \r\n" );
document.write( "\r\n" );
document.write( "                 \"y%2F%28k%5E2x%29\"\"%22%22=%22%22\"\"z%2Fy\"\r\n" );
document.write( "\r\n" );
document.write( "So it is a geometric progression if (but only if) a = k²  \r\n" );
document.write( "                \r\n" );
document.write( "Edwin

\n" ); document.write( " \n" ); document.write( "
\n" );