<br> 设A<sub>k</sub>表示第k次试验成功(k=1,2,…),P{A<sub>k</sub>}=P<sub>k</sub>。<br> P{A<sub>k</sub>}=P{A<sub>k</sub>(A<sub>k</sub><sub>-</sub><sub>1</sub>∪<ruby>A<rp>(</rp><rt style="font-size:7px;layout-grid-mode:line">_</rt><rp>)</rp></ruby><sub>k</sub><sub>-</sub><sub>1</sub>)}=P{A<sub>k</sub>A<sub>k</sub><sub>-</sub><sub>1</sub>}+P{A<sub>k</sub><ruby>A<rp>(</rp><rt style="font-size:7px;layout-grid-mode:line">_</rt><rp>)</rp></ruby><sub>k</sub><sub>-</sub><sub>1</sub>}=P{A<sub>k</sub>,A<sub>k</sub><sub>-</sub><sub>1</sub>}P{A<sub>k</sub><sub>-</sub><sub>1</sub>}+P{A<sub>k</sub><ruby>A<rp>(</rp><rt style="font-size:7px;layout-grid-mode:line">_</rt><rp>)</rp></ruby><sub>k</sub><sub>-</sub><sub>1</sub>}P{<ruby>A<rp>(</rp><rt style="font-size:7px;layout-grid-mode:line">_</rt><rp>)</rp></ruby><sub>k</sub><sub>-</sub><sub>1</sub>}=P{A<sub>k</sub><sub>-</sub><sub>1</sub>}/2+3P{<ruby>A<rp>(</rp><rt style="font-size:7px;layout-grid-mode:line">_</rt><rp>)</rp></ruby><sub>k</sub><sub>-</sub><sub>1</sub>}/4=P{A<sub>k</sub><sub>-</sub><sub>1</sub>}/2+3[1-P{A<sub>k</sub><sub>-</sub><sub>1</sub>}]/4=3/4-P{A<sub>k</sub><sub>-</sub><sub>1</sub>}/4。<br> 所以P{A<sub>n</sub>}=3/4-P{A<sub>n</sub><sub>-</sub><sub>1</sub>}/4(n≥2),即P<sub>n</sub>=3/4-P<sub>n</sub><sub>-</sub><sub>1</sub>/4;<br> (-1/4)P<sub>n</sub><sub>-</sub><sub>1</sub>=(3/4)(-1/4)+(-1/4)<sup>2</sup>P<sub>n</sub><sub>-</sub><sub>2</sub>;<br> (-1/4)<sup>2</sup>P<sub>n</sub><sub>-</sub><sub>2</sub>=(3/4)(-1/4)<sup>2</sup>+(-1/4)<sup>3</sup>P<sub>n</sub><sub>-</sub><sub>3</sub>;<br> (-1/4)<sup>n</sup><sup>-</sup><sup>2</sup>P<sub>2</sub>=(3/4)(-1/4)<sup>n</sup><sup>-</sup><sup>2</sup>+(-1/4)<sup>n</sup><sup>-</sup><sup>1</sup>P<sub>1</sub>。<br> 代入求得<br><span style="display:block;text-align:center;"><img width="512" height="204" src="/s/tiw/p3/UpLoadImage/2019-12-02/001fd6f1-03b2-4927-8226-9bc18b36aa09.png"></span>
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