Question

S =1+3+3^2+3^3+...+3^20

SÓA SÁNH S VỚI 1/2 .3^21

 

Answer 1

\(S=1+3+3^1+3^2+3^3+.....+3^{20}\)

\(3S=3.\left(1+3+3^1+3^2+3^3+.....+3^{20}\right)\)

\(3S=3.1+3.3^1+3.3^2+3.3^3+.....+3.3^{20}\)

\(3S=3+3^2+3^3+3^4+...+3^{21}\)

\(2S=3S-S\)

\(2S=\left(3+3^2+3^3+3^4+.....+3^{21}\right)-\left(1+3^1+3^2+3^3+.....+3^{20}\right)\)

\(2S=3^{21}-1\)

\(\Rightarrow S=\frac{3^{21}-1}{2}\)

\(\frac{1}{2}.3^{21}=3^{21}\div2\)

Vì \(\frac{3^{21}-1}{2}< 3^{21}\div2\)nên S < \(\frac{1}{2}.3^{21}\)