Câu hỏi của Biêtdongsaigon - Toán lớp 6 - Học toán với OnlineMath

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\(\text{Ta có:}\)

\(200-\left(3+\frac{2}{3}+\frac{2}{4}+...+\frac{2}{100}\right)\)

\(\rightarrow200-2-\left(1+\frac{2}{3}+...+\frac{2}{100}\right)\)

\(\rightarrow198-\left(1+\frac{2}{3}+...+\frac{2}{100}\right)\)

\(\rightarrow198-\left(1+\frac{2}{3}+...+\frac{2}{100}\right)\)

\(\rightarrow2.[99-\left(\frac{1}{2}-\frac{1}{3}+...+\frac{1}{100}\right)]\)     \(\left(1\right)\)

\(\text{Ta có:}\)

\(\frac{1}{2}+\frac{2}{3}+...+\frac{99}{100}\)

\(\text{Rút}\)\(\left(1\right)\)\(\text{ra có 99 số}\)

\(\rightarrow99-\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{100}\right)\)     \(\left(2\right)\)

\(\text{Từ}\)\(\left(1\right)\)\(\text{và}\)\(\left(2\right)\)\(\Rightarrow\)\(200-\left(3+\frac{2}{3}+\frac{2}{4}+\frac{2}{5}+...+\frac{2}{100}\right):\left(\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+...+\frac{99}{100}\right)=2\)

A= \(\dfrac{1}{3}-\dfrac{2}{3^2}+....+\dfrac{99}{3^{99}}-\dfrac{100}{3^{100}}\)

3A= 1 - \(\dfrac{2}{3}+\dfrac{3}{3^2}-\dfrac{4}{3^3}+.....+\dfrac{99}{3^{98}}\) - \(\dfrac{100}{3^{99}}\)

A + 3A = 1- \(\dfrac{1}{3}+\dfrac{1}{3^2}\) - \(\dfrac{1}{3^3}+....+\dfrac{1}{3^{99}}-\dfrac{1}{3^{100}}\)

=> 4A < 1 - \(\dfrac{1}{3}+\dfrac{1}{3^2}\) \(\dfrac{1}{3^3}+...+\dfrac{1}{3^{98}}-\dfrac{1}{3^{99}}\)

Đặt : B = 1 - \(\dfrac{1}{3}+\dfrac{1}{3^2}-\dfrac{1}{3^3}+....+\dfrac{1}{3^{98}}-\dfrac{1}{3^{99}}\)

3B = 3 - 1 + \(\dfrac{1}{3}\) - \(\dfrac{1}{3^2}+.....+\dfrac{1}{3^{97}}-\dfrac{1}{3^{98}}\)

B + 3B = 3 - \(\dfrac{1}{3^{99}}\)

4B = 3 - \(\dfrac{1}{3^{99}}\) < 3 => B < \(\dfrac{3}{4}\)

=> 4A < \(\dfrac{3}{4}\) => A < \(\dfrac{3}{16}\) ĐPCM

Ta có : \(\frac{1}{2}+\frac{2}{3}+..+\frac{99}{100}\)

= \((1-\frac{1}{2})+(1-\frac{1}{3})+...+(1-\frac{99}{100})\)(100 cặp số )

= \(\left(1+1+1+...+1\right)-\left(\frac{1}{2}+\frac{1}{3}+...+\frac{1}{100}\right)\)(100 số hạng 1)

= \(1\times100-\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+..+\frac{1}{100}\right)\)

= \(100-\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{100}\right)\)

=> 100-(1+1/2+1/3+...+1/100) = 1/2+2/3+3/4+...+99/100

Bạn cố giải cho mình dễ hiểu hơn ko?

nhanh cho 1 tcik

Ta có   \(A=\frac{200-\left(3+\frac{2}{3}+\frac{2}{4}+\frac{2}{5}+....+\frac{2}{100}\right)}{\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+......+\frac{99}{100}}\)

\(A=\frac{200-2\left(\frac{3}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+....+\frac{1}{100}\right)}{\left(1-\frac{1}{2}\right)+\left(1-\frac{1}{3}\right)+\left(1-\frac{1}{4}\right)+...+\left(1-\frac{1}{100}\right)}\)

\(A=\frac{2\left[100-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+.....+\frac{1}{100}\right)\right]}{100-\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+.....+\frac{1}{100}\right)}\)

\(\Rightarrow A=2\)

Ủa sao bạn ra được \(\frac{200-2\left(\frac{3}{2}+\frac{1}{3}+...+\frac{2}{100}\right)}{\frac{1}{2}+\frac{2}{3}+...+\frac{99}{100}}\)  số 2 ở 200 đâu ra vậy ! và \(\frac{3}{2}\)nữa !

\(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{99.100}< \frac{1}{2}\)

\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{99}-\frac{1}{100}< \frac{1}{2}\)

\(=\frac{1}{2}-\frac{1}{100}< \frac{1}{2}\left(đpcm\right)\)