\(A=5^3+5^4+5^5+...+5^{100}\)
\(A=5^3\left(1+5^1+5^2+...+5^{97}\right)\)
\(A=5^3.\dfrac{5^{97+1}-1}{5-1}=\dfrac{5^3}{4}.\left(5^{98}-1\right)\)
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A=53+54+...+5100
=>5A=54+55+...+5101
=>5A-A=4A=(54+55+...+5101)-(53+54+...+5100)=5101-53
=>A=\(\dfrac{5^{101}-5^3}{4}\)
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