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Arithemetic Basics
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June 14, 2021 8:46 am
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Staff 
(@armor)
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For a natural number n, let t_{n}  be the sum of all digits in  n, for instance,  t_{2009}=2+0+0+9=11,   evaluate

t_{1}+t_{2}+.....+t_{2009}.

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June 14, 2021 8:53 am
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Staff 
(@armor)
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Let   \small T=t_{1}+t_{2}+.......+(2+0+0+8)+(2+0+0+9),   and then reverse the order of right hand side to obtain 

\small T=(2+0+0+9)+(2+0+0+8)+.....+2+1.

Add up these two equalities to obtain 

\small 2T=[1+(2+0+0+9)]+[2+(2+0+0+8)]+......+[(2+0+0+8)+2]+[(2+0+0+9)+1]

\small =12 *2009\Rightarrow T=12*2009/2=12054.

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