Tính nhanh giá trị của các biểu thức sau:
a) \(A=85^2+75^2+65^2+55^2-45^2-35^2-25^2-15^2\)
b) \(B=1^2-2^2+3^2-4^2+5^2-6^2+...+2011^2-2012^2\)
c) \(C=\dfrac{1}{1975}\left(\dfrac{2}{1945}-1\right)-\dfrac{1}{1945}\left(1-\dfrac{2}{1975}\right)-\dfrac{1974}{1975}.\dfrac{1946}{1945}-\dfrac{3}{1975.1945}\)
d) \(D=\left(2^9+2^7+1\right)\left(2^{23}-2^{21}+2^{19}-2^{17}+2^{14}-2^{10}+2^9-2^7+1\right)\)
a) \(A=85^2-45^2+75^2-35^2+65^2-25^2+55^2-15^2\)
\(A=\left(85-45\right)\left(85+45\right)+....+\left(55-15\right)\left(55+15\right)\)
\(A=40.130+40.110+40.90+40.70\)
\(A=40.\left(130+110+90+70\right)=40.400=16000\)
b) \(B=\left(1-2\right)\left(1+2\right)+\left(3-4\right)\left(3+4\right)+...+\left(2011-2012\right)\left(2011+2012\right)\)
\(B=-3-7-11-...-4023\)
\(B=-\left(3+7+11+...+4023\right)\)
\(B=-\dfrac{\left(3+4023\right)\left[\dfrac{\left(4023-3\right)}{4}+1\right]}{2}=2025078\)
\(C=\dfrac{2}{1975.1945}-\dfrac{1}{1975}-\dfrac{1}{1945}+\dfrac{2}{1975.1945}-\dfrac{1974}{1975}.\dfrac{1946}{1945}-\dfrac{3}{1974.1945}\)
\(C=\dfrac{1}{1975.1945}-\dfrac{1}{1975}-\dfrac{1}{1945}-\dfrac{\left(1975-1\right)}{1975}.\dfrac{\left(1945+1\right)}{1945}\)
\(C=\dfrac{1-1945-1975-\left(1975-1\right)\left(1945+1\right)}{1975.1945}\)
\(C=\dfrac{1-1945-1975-\left(1975.1945+1975-1945-1\right)}{1975.1945}\)
\(C=\dfrac{-2.1975-1975.1945+2}{1975.1945}\)
\(C=\dfrac{-2.1974-1975.1945}{1975.1945}=-1-\dfrac{3948}{1975.1945}\)
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