document.write( "Question 79552: Count the number of dots in each arrangement. How many dots will be in the sixth triangular number? . ... ...... \n" ); document.write( "
Algebra.Com's Answer #57113 by Edwin McCravy(20063)\"\" \"About 
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document.write( "Count the number of dots in each arrangement.\r\n" );
document.write( "How many dots will be in the sixth triangular \r\n" );
document.write( "number? . ... ......\r\n" );
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document.write( "Here is why they are called \"triangular numbers\". \r\n" );
document.write( "They are the numbers of dots it takes to make \r\n" );
document.write( "successively bigger triangular arrays of dots.\r\n" );
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document.write( "                                        .\r\n" );
document.write( "                            .          . .\r\n" );
document.write( "                  .        . .        . . .\r\n" );
document.write( "          .      . .      . . .      . . . .\r\n" );
document.write( "    .    . .    . . .    . . . .    . . . . .\r\n" );
document.write( ".  . .  . . .  . . . .  . . . . .  . . . . . .\r\n" );
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document.write( "We can cheat and count the number of dots in the\r\n" );
document.write( "sixth array of dots.  However that's not what your\r\n" );
document.write( "teacher had in mind.  You are to take the first\r\n" );
document.write( "three of 1,3,6 dots and figure out a formula \r\n" );
document.write( "from just those:\r\n" );
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document.write( "The 1st triangle above has\r\n" );
document.write( "1 dot in the top row and\r\n" );
document.write( "that's all there is.  \r\n" );
document.write( "So the first triangular number is 1.\r\n" );
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document.write( "The 2nd triangle above has\r\n" );
document.write( "1 dot in the 1st row and\r\n" );
document.write( "2 dots in the 2nd row.\r\n" );
document.write( "That's 1+2 or 3\r\n" );
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document.write( "The 3rd triangle above has\r\n" );
document.write( "1 dot in the 1st row,\r\n" );
document.write( "2 dots in the 2nd row, and\r\n" );
document.write( "3 dots in the 3rd row.\r\n" );
document.write( "That's 1+2+3 or 6 dots.\r\n" );
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document.write( "So we can see that the nth term\r\n" );
document.write( "is the sum of the arithmetic\r\n" );
document.write( "series 1+2+3+ ··· +n\r\n" );
document.write( "where the difference d is 1.\r\n" );
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document.write( "The formula for the sum of an arithmmtic\r\n" );
document.write( "series is:\r\n" );
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document.write( "Sn = \"n%2F2\"(a1 + an)\r\n" );
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document.write( "where n=6, a1 = 1, an = n\r\n" );
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document.write( "so the formula for the nth triangular number is\r\n" );
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document.write( "Sn = \"n%2F2\"(1 + n)\r\n" );
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document.write( "So when n = 6:\r\n" );
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document.write( "S6 = \"6%2F2\"(1 + 6) = 3(7) = 21\r\n" );
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document.write( "Now count the dots in the last one and we see that\r\n" );
document.write( "it does contain 21 dots.\r\n" );
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document.write( "Edwin
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