Question

CMR: 1/2.2 +1/3.3+1/4.4+....+1/100.100<1

     Mik cần gấp

Answer 1

Ta có : \(\frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+....+\frac{1}{100.100}< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}\)

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

Answer 2

+)Ta thấy:\(\frac{1}{2.2}< \frac{1}{1.2}\)

                   \(\frac{1}{3.3}< \frac{1}{2.3}\)

                     ............................

                     ..............................

                  \(\frac{1}{100.100}< \frac{1}{99.100}\)

\(\Rightarrow\frac{1}{2.2}+\frac{1}{3.3}+...............+\frac{1}{100.100}< \frac{1}{1.2}+\frac{1}{2.3}+............+\frac{1}{99.100}\)

\(\Rightarrow\frac{1}{2.2}+\frac{1}{3.3}+...............+\frac{1}{100.100}< \frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+..............+\frac{1}{99}-\frac{1}{100}< 1\)

\(\Rightarrow\frac{1}{2.2}+\frac{1}{3.3}+.............+\frac{1}{100.100}< 1\left(\text{Đ}PCM\right)\)

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