document.write( "Question 836133: (4/1.2.3)+(5/2.3.4)+(6/3.4.5)+.......
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Algebra.Com's Answer #504117 by Edwin McCravy(20056)\"\" \"About 
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document.write( "The series is\r\n" );
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document.write( "\"sum%28%28k%2B3%29%2F%28k%28k%2B1%29%28k%2B2%29%29%2Ck=1%2Cn%29\"\"%22%22=%22%22\"\"sum%28k%2F%28k%28k%2B1%29%28k%2B2%29%29%2Ck=1%2Cn%29\"\"%22%22%2B%22%22\"\"sum%283%2F%28k%28k%2B1%29%28k%2B2%29%29%2Ck=1%2Cn%29\"\"%22%22=%22%22\"\r\n" );
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document.write( "\"sum%281%2F%28%28k%2B1%29%28k%2B2%29%29%2Ck=1%2Cn%29\"\"%22%22%2B%22%22\"\"3sum%281%2F%28k%28k%2B1%29%28k%2B2%29%29%2Ck=1%2Cn%29\"\r\n" );
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document.write( "Let A(n) = \"sum%281%2F%28%28k%2B1%29%28k%2B2%29%29%2Ck=1%2Cn%29\" and B(n) = \"sum%281%2F%28k%28k%2B1%29%28k%2B2%29%29%2Ck=1%2Cn%29\"\r\n" );
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document.write( "Then we seek to find A(n) + 3·B(n) \r\n" );
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document.write( "We find summation A(n)\r\n" );
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document.write( "By the method of partial fractions:\r\n" );
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document.write( "\"1%2F%28%28k%2B1%29%28k%2B2%29%29\" = \"1%2F%28k%2B1%29-1%2F%28k%2B2%29\"\r\n" );
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document.write( "A(n) = \"sum%281%2F%28%28k%2B1%29%28k%2B2%29%29%2Ck=1%2Cn%29\"\"%22%22=%22%22\"\"sum%28%281%2F%28k%2B1%29-1%2F%28k%2B2%29%29%2Ck=1%2Cn%29\"\"%22%22=%22%22\"\r\n" );
document.write( "\"%22%22=%22%22\"\"1%2F2-1%2F%28n%2B2%29\"\r\n" );
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document.write( "Now we find B(n):\r\n" );
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document.write( "Also by the method of partial fractions:\r\n" );
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document.write( "\"1%2F%28k%28k%2B1%29%28k%2B2%29%29\"\"%22%22=%22%22\"\"%281%2F2%29%2Fk-1%2F%28k%2B1%29%2B%281%2F2%29%2F%28k%2B2%29\"\"%22%22=%22%22\"\"expr%281%2F2%29%281%2Fk-2%2F%28k%2B1%29%2B1%2F%28k%2B2%29%29\"\"%22%22=%22%22\"\r\n" );
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document.write( "\"expr%281%2F2%29%281%2Fk-1%2F%28k%2B1%29-1%2F%28k%2B1%29%2B1%2F%28k%2B2%29%29\"\"%22%22=%22%22\"\"expr%281%2F2%29%28%281%2Fk-1%2F%28k%2B1%29%29-%281%2F%28k%2B1%29-1%2F%28k%2B2%29%29%29\", so\r\n" );
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document.write( "B(n) = \"sum%281%2F%28k%28k%2B1%29%28k%2B2%29%29%2Ck=1%2Cn%29\"\"%22%22=%22%22\"\"%22%22=%22%22\"\r\n" );
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document.write( "\"%22%22=%22%22\"\r\n" );
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document.write( "(Since that last sum is A(n) or \"1%2F2-1%2F%28n%2B2%29\"\r\n" );
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document.write( "\"%22%22=%22%22\"\r\n" );
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document.write( "And since \"sum%28+%281%2Fk-1%2F%28k%2B1%29%29%2Ck=1%2Cn%29\" = \"1%2F1-1%2F2%2B1%2F2-1%2F3%2B%22...%22%2B1%2F%28n-1%29-1%2Fn%2B1%2Fn-1%2F%28n%2B1%29\"\"%22%22=%22%22\"\"1-1%2F%28n%2B1%29\"\r\n" );
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document.write( "B(n) = \"expr%281%2F2%29%281-1%2F%28n%2B1%29-%281%2F2-1%2F%28n%2B2%29%29%29\"\"%22%22=%22%22\"\"expr%281%2F2%29%281-1%2F%28n%2B1%29-1%2F2%2B1%2F%28n%2B2%29%29\"\"%22%22=%22%22\"\"expr%281%2F2%29%281%2F2-1%2F%28n%2B1%29%2B1%2F%28n%2B2%29%29\"\"%22%22=%22%22\"\"expr%281%2F2%29%281%2F2-1%2F%28n%2B1%29%2B1%2F%28n%2B2%29%29\"\"%22%22=%22%22\"\r\n" );
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document.write( "So:\r\n" );
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document.write( "\"sum%28%28k%2B3%29%2F%28k%28k%2B1%29%28k%2B2%29%29%2Ck=1%2Cn%29\"\"%22%22=%22%22\"A(n)+3B(n)\"%22%22=%22%22\"\"1%2F2-1%2F%28n%2B2%29%2Bexpr%283%2F2%29%281%2F2-1%2F%28n%2B1%29%2B1%2F%28n%2B2%29%29\"\r\n" );
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document.write( "Getting LCDs, combining terms and factoring, the sum becomes:\r\n" );
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document.write( "\"sum%28%28k%2B3%29%2F%28k%28k%2B1%29%28k%2B2%29%29%2Ck=1%2Cn%29\"\"%22%22=%22%22\"\"%28n%2A%285n%2B11%29%29%2F%284%28n%2B1%29%28n%2B2%29%29\"\r\n" );
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document.write( "Edwin
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