document.write( "Question 1175017: Find a formula for the nth triangular number Sn = 1 + 2 + 3 + · · · + n.
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Algebra.Com's Answer #800555 by math_helper(2461) $\"\"$ $\"About$

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document.write( "One way to get a formula is to add 1+2+...+n to itself, but write the 2nd line in reverse, then add them:\r
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document.write( "Sn  =    1 +    2   +   3  + ... + (n-1) + n 
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document.write( "Sn  =  ( n + (n-1) + (n-2) + ... +    2  + 1 )
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document.write( "2Sn = (n+1) + (n+1) + (n+1) + ... + (n+1) + (n+1)\r
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document.write( "The RHS is just n terms of n+1, thus it is n(n+1)\r
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document.write( "2Sn = n(n+1)\r
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document.write( "   \r
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document.write( "(i)
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document.write( "As a hint for divisibility by 5: 
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document.write( "      If n is even, n+1 is odd
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document.write( "      If n is odd, n+1 is even\r
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document.write( "Thus, a factor of 2 is always present in n(n+1), and the (1/2) in the  formula will divide that out (\"cancels\" it).\r
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document.write( "So there is just the consideration about n or n+1 being divisible by 5.  Numbers ending with 5 or 0 meet this condition, and as long as EITHER n or n+1 ends with 5 or 0 you will find  to be divisible by 5:  \r
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document.write( "Examples: 
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document.write( "n=10  >>>  n*(n+1)/2 = 110/2 = 55,  that works.     
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document.write( "n=9   >>>  n*(n+1)/2 = 9*10/2 = 45, that is divisible by 5
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document.write( "n=8   >>>  n*(n+1)/2 = 8*9/2 = 36, NOT divisible by 5 (neither n nor n+1 is divisble by 5)
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document.write( "n=15  >>>  n*(n+1)/2 = 15*16/2 = 120, divisible by 5
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document.write( "n=14  >>>  n*(n+1)/2 = 14*15/2 = 105, divisible by 5\r
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document.write( "A subtle note about the above: notice how the factor of 5 comes from 'n' or 'n+1' in a mutually exclusive way.  That is, if n has a factor of 5, n+1 does not;  if n+1 has a factor of 5, n does not.  You may need this observation for part(ii) when considering the factor 4.\r
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document.write( "(ii)
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document.write( " is even:  you give it some thought.  Remember, you always get one factor of 2 automatically in the product n(n+1) and that will cancel with the 1/2 of the formula, so the question is, for what values of n(n+1) do you get n(n+1) divisible by 4?\r
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