Question

TÍNH : H= \(\frac{1}{5^2}\)+\(\frac{2}{5^3}\)+........+\(\frac{11}{5^{12}}\)

hic. giúp mk vs T .T . 

À thêm 1 tí nhé : Cm H < \(\frac{1}{16}\)

Answer 1

\(\Rightarrow5H=\frac{1}{5}+\frac{2}{5^2}+...+\frac{11}{5^{11}}\)

\(\Rightarrow5H-H=\left(\frac{1}{5}+\frac{2}{5^2}+...+\frac{11}{5^{11}}\right)-\left(\frac{1}{5^2}+\frac{2}{5^3}+...+\frac{11}{5^{12}}\right)\)

\(\Rightarrow4H=\frac{1}{5}+\frac{1}{5^2}+..+\frac{1}{5^{11}}-\frac{11}{5^{12}}\)

Đặt \(A=\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{11}}\)

\(\Rightarrow5A=1+\frac{1}{5}+...+\frac{1}{5^{10}}\)

\(\Rightarrow5A-A=\left(1+..+\frac{1}{5^{10}}\right)-\left(\frac{1}{5}+...+\frac{1}{5^{11}}\right)\)

\(\Rightarrow4A=1-\frac{1}{5^{11}}\)

\(\Rightarrow A=\frac{1}{4}-\frac{1}{4.5^{11}}\)

\(\Rightarrow4H=\frac{1}{4}-\frac{1}{4.5^{11}}-\frac{11}{5^{12}}\)

\(\Rightarrow H=\frac{1}{16}-\frac{1}{4^2.5^{11}}-\frac{11}{4.5^{12}}\)

Answer 2

Ta có : \(5H=\frac{1}{5}+\frac{2}{5^2}+...+\frac{11}{5^{11}}\)

\(\Rightarrow4H=\left(\frac{1}{5}+\frac{2}{5^2}+...+\frac{11}{5^{11}}\right)-\left(\frac{1}{5^2}+\frac{2}{5^3}+...+\frac{11}{5^{12}}\right)=\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{11}}+\frac{11}{5^{12}}\)

\(\Rightarrow20H=1+\frac{1}{5}+...+\frac{1}{5^{10}}+\frac{11}{5^{11}}\)

\(\Rightarrow16H=20H-4H=1+\frac{10}{5^{11}}-\frac{11}{5^{12}}\Leftrightarrow H=\frac{1+\frac{10}{5^{11}}-\frac{11}{5^{12}}}{16}.\)

Answer 3

Đỗ Đức Lợi nhầm rồi 

Phải là \(\frac{1+\frac{1}{5^{11}}-\frac{11}{5^{12}}}{16}\)nhé cu

Answer 4

H = \(\frac{1}{5^2}+\frac{2}{5^3}+...+\frac{11}{5^{12}}\)

5H = \(\frac{1}{5}+\frac{2}{5^2}+...+\frac{11}{5^{11}}\)

5H - H = \(\left(\frac{1}{5}+\frac{2}{5^2}+...+\frac{11}{5^{11}}\right)-\left(\frac{1}{5^2}+\frac{2}{5^3}+...+\frac{11}{5^{12}}\right)\)

4H  = \(\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{11}}-\frac{1}{5^{12}}\)

20H = \(1+\frac{1}{5}+...+\frac{1}{5^{10}}-\frac{1}{5^{11}}\)

20H - 4H = \(\left(1+\frac{1}{5}+...+\frac{1}{5^{10}}-\frac{1}{5^{11}}\right)-\left(\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{11}}-\frac{1}{5^{12}}\right)\)

16H = \(1-\frac{5}{5^{11}}-\frac{11}{5^{11}}+\frac{11}{5^{12}}\)

        = \(1-\left(\frac{5}{5^{12}}+\frac{5^5}{5^{12}}-\frac{11}{5^{12}}\right)\)

         = \(1-\frac{44}{5^{12}}\)

Thấy \(1-\frac{44}{5^{12}}< 1\)=> 16H < 1 => H < \(\frac{1}{16}\)( đpcm )

Answer 5

cảm ơn nhé

Answer 6

:V Tiểu Ngôn đầu bài có bảo CM đâu

Answer 7

#C-TMF ( Team TST 23 ) : Đọc dòng thứ 3 của đầu bài  ._.