document.write( "Question 614717: What is a recursive and an explicit formula for the sequence: 2/3, 4/9, 16/81 ... \n" ); document.write( "
Algebra.Com's Answer #386673 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Wr'll find the explicit formula first:\r\n" );
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document.write( "Sequence 1 = \"2%2F3\", \"4%2F9\", \"16%2F81\", ...\r\n" );
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document.write( "That's the sequence\r\n" );
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document.write( "a1=\"%282%2F3%29%5E1\", a2=\"%282%2F3%29%5E2\", a3=\"%282%2F3%29%5E4\", ...\r\n" );
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document.write( "The exponents follow the sequence\r\n" );
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document.write( "Sequence 2:  1, 2, 4, ...\r\n" );
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document.write( "That's the sequence:\r\n" );
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document.write( "20, 21, 22, ...\r\n" );
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document.write( "Those exponents follow the sequence:\r\n" );
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document.write( "Sequence 3:  0, 1, 2\r\n" );
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document.write( "which follow the sequence\r\n" );
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document.write( "1-1, 2-1, 3-1\r\n" );
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document.write( "which has nth term:\r\n" );
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document.write( "n-1\r\n" );
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document.write( "Therefore sequence 2 has nth term:\r\n" );
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document.write( "2n-1\r\n" );
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document.write( "Therefore sequence 1 has sequence:\r\n" );
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document.write( "an = \"%282%2F3%29%5E%282%5E%28n-1%29%29\" \r\n" );
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document.write( "That's the explicit formula.\r\n" );
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document.write( "---------------------------------\r\n" );
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document.write( "So the sequence is\r\n" );
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document.write( "\"%282%2F3%29%5E%282%5E0%29\", \"%282%2F3%29%5E%282%5E1%29\", \"%282%2F3%29%5E%282%5E2%29\", ...\r\n" );
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document.write( "To find a recursion formula:\r\n" );
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document.write( "Divide the (n+1)st term by the nth term:\r\n" );
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document.write( "\"%282%2F3%29%5E%282%5E%28%28n%2B1%29-1%29%29\"÷\"%282%2F3%29%5E%282%5E%28n-1%29%29\"\r\n" );
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document.write( "\"%282%2F3%29%5E%282%5E%28n%29%29\"÷\"%282%2F3%29%5E%282%5E%28n-1%29%29\"\r\n" );
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document.write( "\"%282%2F3%29%5E%282%5En-2%5E%28n-1%29%29\"\r\n" );
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document.write( "\"%282%2F3%29%5E%282%5E%28n-1%29%282-1%29%29\"\r\n" );
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document.write( "\"%282%2F3%29%5E%282%5E%28n-1%29%281%29%29\"\r\n" );
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document.write( "\"%282%2F3%29%5E%282%5E%28n-1%29%29\"  \r\n" );
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document.write( "an+1 = an·(\"2%2F3\")2n-1\r\n" );
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document.write( "And the complete recursion formula is:\r\n" );
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document.write( "a1 = \"2%2F3\", an+1 = an+1 = an·(\"2%2F3\")2n-1\r\n" );
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document.write( "Edwin
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