Question

[Nguyễn Đức Hưng]

Cho S= 5/3 + 8/3^ 2 + 11/3^ 3 + 14/3^ 4 +....+ 302/3^ 100 CMR S<3½

Answer 1

[Hiền Thương]

\(\Rightarrow3s=5+\frac{8}{3}+\frac{11}{3^2}+\frac{14}{3^3}+...+\frac{302}{3^{99}}\)

\(\Rightarrow3S-S=5+1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{98}}-\frac{302}{3^{100}}\)  (1)

Đặt A = \(1+\frac{1}{3}+\frac{1}{3^2}+....+\frac{1}{3^{98}}\)

\(\Rightarrow3A=3+1+\frac{1}{3}+....+\frac{1}{3^{97}}\)

\(\Rightarrow3A-A=3-\frac{1}{3^{98}}\)

\(\Rightarrow2A=3-\frac{1}{3^{98}}\)

\(\Rightarrow A=\frac{3}{2}-\frac{1}{3^{98}.2}\)

  từ  (1)   \(\Rightarrow2S=\)\(5+\frac{3}{2}-\frac{1}{3^{98}.2}-\frac{302}{3^{100}}\) 

           \(\Rightarrow S=\frac{5}{2}+\frac{3}{2.2}-\frac{1}{3^{98}.4}-\frac{302}{3^{100}.2}\)  

                  \(\Rightarrow S=\frac{13}{4}-\frac{1}{3^{98}.4}-\frac{302}{3^{100}.2}\) 

VÌ \(\frac{13}{4}< \frac{14}{4}\) Hay \(\frac{13}{14}< 3\frac{1}{2}\) 

\(\frac{\Rightarrow13}{4}-\frac{1}{3^{98}.4}-\frac{302}{3^{100}.2}< 3\frac{1}{2}\)

 Vậy S < \(3\frac{1}{2}\)

Answer 2

[Nguyễn Đức Hưng]

Cảm ơn bn nhiều nha