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a) \(2^3+4^3+6^3+...+20^3\)
\(=2^3\left(1+2^3+...+10^3\right)\)
\(=8\cdot3025\)
\(=24200\)
b) \(0,5^3+1^3+1,5^3+...+5^3\)
\(=0,5^3\left(1+2^3+...+10^3\right)\)
\(=0,125\cdot3025\)
\(=378,125\)
S = 23 + 43 + 63 +...+ 103
= (1.2)3 + (2.2)3 + .... + (2.5)3
= 23 .(13 + 23 + 33 +...+ 53)
= 23 . 3025
= 24200
a) \(\frac{3^7.16^3}{12^5.27^2}=\frac{3^7.\left(4^2\right)^3}{3^5.4^5.\left(3^3\right)^2}\)
\(=\frac{3^7.4^6}{3^5.4^5.3^6}\)
\(=\frac{3^7.4^6}{3^{11}.4^5}\)
\(=\frac{4}{3^4}=\frac{4}{81}\)
b) \(\frac{2^3.\left(\frac{1}{2}\right)^3.3^7}{2.\left(\frac{1}{2}\right)^4.3^8}\)
\(=\frac{4}{\frac{1}{2}.3}=\frac{4}{\frac{3}{2}}=\frac{8}{3}\)
c)\(\frac{10^3+5.10^2+5^3}{-13}\)
\(=\frac{10^2.\left(10+5\right)+125}{-13}\)
\(=\frac{100.15+125}{-13}\)
\(=\frac{1500+125}{-13}=\frac{1625}{-13}=-125\)
\(S=\left(0,25\right)^2+\left(0,5\right)^2+...+\left(2,5\right)^2\)
\(\Rightarrow\frac{n\left(n+1\right)\left(2n+1\right)}{6}=\frac{2,5\left(2,5+1\right)\left(2,5.2+1\right)}{6}\)
\(\Rightarrow S=8,75\)
