document.write( "Question 841301: hence,prove algebraically that the sum of any two consecutive terms is a perfect square 3;x;10;y;21 \n" ); document.write( "
Algebra.Com's Answer #507788 by Edwin McCravy(20056) $\"\"$ $\"About$
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document.write( "You start your question with \"hence\", so we cannot be sure\r\n" );
document.write( "what went before that, and how you are supposed to come up\r\n" );
document.write( "with x and y.  I observe that the three numbers given 3,10,21 \r\n" );
document.write( "appear in Pascal's triangle \r\n" );
document.write( "      \r\n" );
document.write( "              1\r\n" );
document.write( "            1   1\r\n" );
document.write( "          1   2   1\r\n" );
document.write( "        1   3    3   1\r\n" );
document.write( "      1   4   6   4   1\r\n" );
document.write( "    1   5   10  10   5   1\r\n" );
document.write( "  1   6  15  20  15   6   1\r\n" );
document.write( "1   7   21  35  35  21   7   1\r\n" );
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document.write( "Pascal's triangle is composed of binomial coefficients which \r\n" );
document.write( "are combinations: \r\n" );
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document.write( "3 = C(3,2), 10 = C(5,2), 21 = C(7,2)\r\n" );
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document.write( "So I assume that the sequence 3,x,10,y,21 is this sequence:\r\n" );
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document.write( "3=C(3,2), x=C(4,2), 10=C(5,2), y=C(6,2), 21=C(7,2) \r\n" );
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document.write( ",,,,\r\n" );
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document.write( "So the nth term is , and they are called\r\n" );
document.write( "\"triangular\" numbers, because they are the numbers of dots \r\n" );
document.write( "that can be formed into a triangular arraylike this:                                                         .\r\n" );
document.write( "                                              .               . .\r\n" );
document.write( "                              .              . .             . . .\r\n" );
document.write( "                .            . .            . . .           . . . .\r\n" );
document.write( "     .         . .          . . .          . . . .         . . . . .\r\n" );
document.write( "3 = . .   6 = . . .,  10 = . . . .,  15 = . . . . ., 21 = . . . . . .\r\n" );
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document.write( "We need to prove that the sum of any two consecutive terms of this sequence\r\n" );
document.write( "3,6,10,15,21,... is a perfect square.  We can see that 3+6=9=3², 6+10=16=4²,\r\n" );
document.write( "10+15=25=5², 15+21=36=6².  We need to prove this in general:\r\n" );
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document.write( "Proof:\r\n" );
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document.write( "the nth term is  and \r\n" );
document.write( "the (n+1)st term is  = \r\n" );
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document.write( "Add them:\r\n" );
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document.write( "which is a perfect square.\r\n" );
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document.write( "Edwin
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