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Let {F(n)} = {1,1,2,3,5,8,13,21,34,55,···} be the Fibonacci sequence\r\n" );
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document.write( "F(1) = F(2) = 1, F(n)= F(n-1) + F(n-2) if n > 2. \r\n" );
document.write( "Show that holds for n that is greator or equal to 1.\r\n" );
document.write( "F(2) + F(4)+....+ F(2n) = F(2n+1) - 1\r\n" );
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document.write( "Proof by induction:\r\n" );
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document.write( "It holds for n = 2 since F(2) + F(4) = 4 and F(2·2+1) - 1 = F(4+1) - 1 =\r\n" );
document.write( "F(5) - 1 = 5 - 1 = 4 \r\n" );
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document.write( "Assume that it holds for some n = k ³ 2.\r\n" );
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document.write( "That is, \r\n" );
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document.write( "F(2) + F(4) + ··· + F(2k) = F(2k+1) - 1\r\n" );
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document.write( "We need to show that under this assumption it also holds for n = k+1.\r\n" );
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document.write( "That is, we need to show that\r\n" );
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document.write( "F(2) + F(4) + ··· + F( 2(k+1) ) = F( 2(k+1)+1 ) - 1 = F(2k+3) - 1\r\n" );
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document.write( "By induction hypothesis, the left side equals to\r\n" );
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document.write( "F(2k+1)-1 + F( 2(k+1) ) = F(2k+1) + F(2k+2) - 1 which by definition\r\n" );
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document.write( "equals F(2k+3) - 1, which is what we had to prove.\r\n" );
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document.write( "QED\r\n" );
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document.write( "Edwin\r\n" );
document.write( "AnlytcPhil@aol.com\n" ); document.write( " \r
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