document.write( "Question 1209312: In the sequence 1, 3, 6, 10, 15, 21, ..., in the first 1000 terms of the sequence, how many terms are multiples of 5? \n" ); document.write( "

Algebra.Com's Answer #848204 by ikleyn(52754) $\"\"$ $\"About$

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\n" ); document.write( "In the sequence 1, 3, 6, 10, 15, 21, ..., in the first 1000 terms of the sequence,
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document.write( "The n-th term of this sequence,  ,  is the sum of the first n natural numbers\r\n" );
document.write( "     = 1,\r\n" );
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document.write( "     = 3 = 1 + 2,\r\n" );
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document.write( "     = 6 = 1 + 2 + 3,\r\n" );
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document.write( "     = 10 = 1 + 2 + 3 + 4,\r\n" );
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document.write( "     = 15 = 1 + 2 + 3 + 4 + 5,\r\n" );
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document.write( "and so on.  For the sum of the first natural numbers, there is the formula\r\n" );
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document.write( "     = .\r\n" );
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document.write( "So, the question is: how many terms    of the first 1000 terms are multiples of 5.\r\n" );
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document.write( "In order for the term    be a multiple of 5, EITHER  n  OR  (n+1)  must be multiple of 5\r\n" );
document.write( "(here \"either - or\"  is exclusive).\r\n" );
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document.write( "Of the first 1000 numbers n = 1, 2, 3, 4, 5, . . . , 1000,  there are  1000/5 = 200 multiple of 5.\r\n" );
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document.write( "Of the first 1000 numbers n+1 = 2, 3, 4, 5, . . . , 1001,  there are   = 200 other numbers multiple of 5.\r\n" );
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document.write( "Thus, in all,  among first 1000 numbers   =  ,  there are  200 + 200 = 400 numbers multiple of 5.\r\n" );
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document.write( "ANSWER.  In the sequence 1, 3, 6, 10, 15, 21, ..., in the first 1000 terms of the sequence, \r\n" );
document.write( "         400 numbers are multiple of 5.\r\n" );
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