补写出下列句子中的空缺部分。

如图所示，BC为半径R=2.5m的 $\frac{1}{4}$ 圆弧，AB为光滑水平轨道，两轨道在B处相切连接；AB轨道上的滑块P通过不伸长的轻绳与套在竖直光滑细杆的滑块Q连接；开始时，P在A处，Q在与A同一水平面上的E处，且绳子刚好伸直处于水平，固定的小滑轮在D处，DE=3m不计滑轮与绳子间的摩擦和空气阻力，现把Q从静止释放，当下落h=3m时，P恰好到达圆弧轨道的B，且对B无压力．取g=10m/s²．试求：

（Ⅰ）求证：数列{a_n}为等差数列；

（Ⅱ）试求所有的正整数m，使得 $\frac{{\text{a}}_{\text{m}}{}^{\text{2}}{\text{+a}}_{\text{m+1}}{}^{\text{2}}-{\text{a}}_{\text{m+2}}{}^{\text{2}}}{{\text{a}}_{\text{m}}{\text{a}}_{\text{m+1}}}$ 为整数；

（Ⅲ）若对任意的n∈N^*，不等式 $\lambda {\text{T}}_{\text{n}}\text{<n+}\frac{\text{2}}{\text{3}}•{\left(-\text{1}\right)}^{\text{n+1}}$ 恒成立，求实数λ的取值范围．

      Ruby, the artist, is an Asian elephant, She is17 years old. She lives in the zoo. Ruby has produced about 100 paintings since last year. She has made more than $ 40,000 for the zoo. Zoo leaders say Ruby first showed the talent (才能) about four years ago. Zoo workers watched her make the marks in the sand with her trunk. She rubbed (擦掉) out the marks, then made the new ones. The zoo worker gave her brushes and paints. Soon Ruby painted pictures.
       Newspaper reporters wrote Ruby, then people asked about buying her pictures. And in this way, she made a lot of money for the zoo. This is how Ruby does the paintings. A zoo worker holds different color paints. Ruby touches the colors she wants. Then the worker gives her brushes with the chosen colors in her trunk and Ruby begins painting.