Ta có:
a) \(S=2^3+2^5+2^7+...+2^{25}\)
\(\Rightarrow2^2\cdot S=2^2\cdot\left(2^3+2^5+2^7+...+2^{25}\right)\)
\(\Rightarrow4\cdot S=2^5+2^7+2^9+...+2^{27}\)
\(\Rightarrow4\cdot S-S=\left(2^5+2^7+2^9+...+2^{27}\right)-\left(2^3+2^5+2^7+...+2^{25}\right)\)
\(\Rightarrow3\cdot S=2^{27}-2^3\)
\(\Rightarrow S=\frac{2^{27}-2^3}{3}\)
b) \(S=3+3^2+3^3+...+3^{100}\)
\(\Rightarrow3\cdot S=3\cdot\left(3+3^2+3^3+...+3^{100}\right)\)
\(\Rightarrow3\cdot S=3^2+3^3+3^4+...+3^{101}\)
\(\Rightarrow3\cdot S-S=\left(3^2+3^3+3^4+...+3^{101}\right)-\left(3+3^2+3^3+...+3^{100}\right)\)
\(\Rightarrow2\cdot S=3^{101}-3\)
\(\Rightarrow S=\frac{3^{101}-3}{2}\)
