document.write( "Question 862408: Prove by mathematical induction\r
\n" ); document.write( "\n" ); document.write( "\"1%2F%281%2A2%29+%2B+1%2F%282%2A3%29+%2B+1%2F%283%2A4%29\"+ ...+\"1%2Fn%28n%2B1%29+=+n%2F%28n%2B1%29\"\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #519657 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "\"1%2F%281%2A2%29+%2B+1%2F%282%2A3%29+%2B+1%2F%283%2A4%29%2B%22...%22%2B1%2F%28n%28n%2B1%29%29\"\"%22%22=%22%22\"\"n%2F%28n%2B1%29\"\r\n" );
document.write( "\r\n" );
document.write( "We substitute n=1 \r\n" );
document.write( "\r\n" );
document.write( "\"1%2F%281%2A2%29\"\"%22%22=%22%22\"\"1%2F%281%2B1%29\"\r\n" );
document.write( "\r\n" );
document.write( "That's true.\r\n" );
document.write( "\r\n" );
document.write( "Assume true for n=k\r\n" );
document.write( "\r\n" );
document.write( "(1)  \"1%2F%281%2A2%29+%2B+1%2F%282%2A3%29+%2B+1%2F%283%2A4%29%2B%22...%22%2B1%2F%28k%28k%2B1%29%29\"\"%22%22=%22%22\"\"k%2F%28k%2B1%29\"\r\n" );
document.write( "\r\n" );
document.write( "We want to show that if (1) is true, then the following is true:\r\n" );
document.write( "\r\n" );
document.write( "     \"%22%22=%22%22\"\"%28k%2B1%29%2F%28%28k%2B1%29%2B1%29\"\r\n" );
document.write( "\r\n" );
document.write( "which simplifies to this:\r\n" );
document.write( "\r\n" );
document.write( "(2)  \"%22%22=%22%22\"\"%28k%2B1%29%2F%28k%2B2%29\"\r\n" );
document.write( "\r\n" );
document.write( "----------------------------------------------\r\n" );
document.write( "\r\n" );
document.write( "So we start with what we assume to be true, which is (1):\r\n" );
document.write( "\r\n" );
document.write( "\"1%2F%281%2A2%29+%2B+1%2F%282%2A3%29+%2B+1%2F%283%2A4%29%2B%22...%22%2B1%2F%28k%28k%2B1%29%29\"\"%22%22=%22%22\"\"k%2F%28k%2B1%29\"\r\n" );
document.write( "\r\n" );
document.write( "and add the next term \"red%281%2F%28%28k%2B1%29%28k%2B2%29%29%29\" to both sides:\r\n" );
document.write( "\r\n" );
document.write( "\"%22%22=%22%22\"\"k%2F%28k%2B1%29%2Bred%281%2F%28%28k%2B1%29%28k%2B2%29%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "Now if we can show that the right side\r\n" );
document.write( "\r\n" );
document.write( "\"k%2F%28k%2B1%29%2B1%2F%28%28k%2B1%29%28k%2B2%29%29\"\r\n" );
document.write( "\r\n" );
document.write( "is equal to the right side of (2) then we will have proved the\r\n" );
document.write( "proposition by induction:\r\n" );
document.write( "\r\n" );
document.write( "\"k%2F%28k%2B1%29%2B1%2F%28%28k%2B1%29%28k%2B2%29%29\" =\r\n" );
document.write( "\r\n" );
document.write( "We get an LCD of (k+1)(k+2)\r\n" );
document.write( "\r\n" );
document.write( "\"%28k%28k%2B2%29%29%2F%28%28k%2B1%29%28k%2B2%29%29%2B1%2F%28%28k%2B1%29%28k%2B2%29%29\" =\r\n" );
document.write( "\r\n" );
document.write( "\"%28k%28k%2B2%29%2B1%29%2F%28%28k%2B1%29%28k%2B2%29%29\" =\r\n" );
document.write( "\r\n" );
document.write( "\"%28k%5E2%2B2k%2B1%29%2F%28%28k%2B1%29%28k%2B2%29%29\" =\r\n" );
document.write( "\r\n" );
document.write( "\"%28%28k%2B1%29%28k%2B1%29%29%2F%28%28k%2B1%29%28k%2B2%29%29\" =\r\n" );
document.write( "\r\n" );
document.write( "\"%28%28cross%28k%2B1%29%29%28k%2B1%29%29%2F%28%28cross%28k%2B1%29%29%28k%2B2%29%29\" =\r\n" );
document.write( "\r\n" );
document.write( "\"%28k%2B1%29%2F%28k%2B2%29\"\r\n" );
document.write( "\r\n" );
document.write( "That's the same as the right side of (2) above.  So we are done.\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );