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Find the exact value of 2012-2002+1992-1982 +...+ 32 - 22 + 12 - 2

 May 7, 2021
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Find the exact value of 2012-2002+1992-1982 +...+ 32 - 22 + 12 - 2

 

\(\begin{array}{|l|rcl|} \hline 1.& -2+12 &=& 10 \\ 2.& -22+32 &=& 10 \\ 3.& -42+52 &=& 10 \\ \ldots & \ldots &\ldots & \ldots \\ n-1 & -1982+1992 &=& 10 \\ n & -2002+2012 &=& 10 \\ \hline & sum &=& 10*n \\ & sum &=& 10*101 \\ & \mathbf{sum} &=& \mathbf{1010} \\ \hline \end{array} \begin{array}{|rcll|} \hline &&\text{ finite arithmetic progression} \\ \hline a_n &=& a_1+(n-1)d \quad | \quad a_n = 2012,~a_1 = 12,~ d = 20 \\ 2012 &=& 12+(n-1)*20 \\ 2000 &=& (n-1)*20 \\ 100 &=& n-1 \\ \mathbf{n} &=& \mathbf{101} \\ \hline \end{array}\)

 

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 May 7, 2021

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