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# functional operations

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For positive integer values of N, let boxed(N) be equal to:

boxed(N) = 2+4+6+8+...+N if N is even,

and

boxed(N) = 1+3+5+7+...+N if N is odd.

What is the value of boxed(2009) minus boxed(2008)?

May 23, 2022
edited by idontknowhowtodivide  May 23, 2022

#1
-2

boxed(2009) - boxed(2008) = (1 + 3 + 5 + ... + 2009) - (2 + 4 + ... + 2008) = 504.

May 23, 2022
#2
+26383
+3

For positive integer values of N, let boxed(N) be equal to:

boxed(N) = 2+4+6+8+...+N if N is even,
and
boxed(N) = 1+3+5+7+...+N if N is odd.

What is the value of boxed(2009) minus boxed(2008)?

$$\begin{array}{|rcll|} \hline s &=& boxed(2009) \quad minus \quad boxed(2008) \\ s &=& (1 + 3 + 5 + ... + 2009) - (2 + 4 + ... + 2008) \\ s &=& 1 - 2 + 3 - 4 + \dots + 2007 - 2008 + 2009 \\ \hline \end{array}$$

$$\begin{array}{|rcll|} \hline s &=& 1 - 2 + 3 - 4 + \dots + 2007 - 2008 + 2009 \\ s &=& 2009 - 2008 +2007-2006 + \dots +3-2+1 \\ \hline 2s &=& 2010-2010+2010-2010 + \dots + 2010-2010+2010 \\ 2s &=& 0+0+ \dots +0+2010 \\ 2s &=& 2010 \quad | \quad : 2\\ s &=& 1005 \\ \hline \end{array}$$

boxed(2009) minus boxed(2008)  = $$\mathbf{1005}$$

May 24, 2022
#3
+186
+3

Thanks heureka for the help!

idontknowhowtodivide  May 24, 2022