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Question Number 157891 by HongKing last updated on 29/Oct/21

Answered by TheSupreme last updated on 29/Oct/21

((3/4)+(5/(36))+(7/(144))+(9/(400))+((11)/(900)))x<70 (((2400+500+175+81+44)/(3600)))x<70 ((8/9))x<70 x<((630)/8)=78.75 largest x = 78

$$\left(\frac{\mathrm{3}}{\mathrm{4}}+\frac{\mathrm{5}}{\mathrm{36}}+\frac{\mathrm{7}}{\mathrm{144}}+\frac{\mathrm{9}}{\mathrm{400}}+\frac{\mathrm{11}}{\mathrm{900}}\right){x}<\mathrm{70} \\ $$$$\left(\frac{\mathrm{2400}+\mathrm{500}+\mathrm{175}+\mathrm{81}+\mathrm{44}}{\mathrm{3600}}\right){x}<\mathrm{70} \\ $$$$\left(\frac{\mathrm{8}}{\mathrm{9}}\right){x}<\mathrm{70} \\ $$$${x}<\frac{\mathrm{630}}{\mathrm{8}}=\mathrm{78}.\mathrm{75} \\ $$$${largest}\:{x}\:=\:\mathrm{78} \\ $$$$ \\ $$$$ \\ $$

Commented by HongKing last updated on 29/Oct/21

a)74 b)71 c)69 d)64 e)55

$$\left.\mathrm{a}\left.\right)\left.\mathrm{7}\left.\mathrm{4}\left.\:\:\:\mathrm{b}\right)\mathrm{71}\:\:\:\mathrm{c}\right)\mathrm{69}\:\:\:\mathrm{d}\right)\mathrm{64}\:\:\:\mathrm{e}\right)\mathrm{55} \\ $$

Answered by mr W last updated on 29/Oct/21

((2k+1)/(k^2 (k+1)^2 ))=(1/k^2 )−(1/((k+1)^2 )) Σ=((1/1^2 )−(1/2^2 ))+((1/2^2 )−(1/3^2 ))+...+((1/5^2 )−(1/6^2 )) =(1/1^2 )−(1/6^2 )=1−(1/(36))=((35)/(36)) ((35)/(36))x<70 x<((70×36)/(35))=72 ⇒x_(max) =71 ⇒answer b

$$\frac{\mathrm{2}{k}+\mathrm{1}}{{k}^{\mathrm{2}} \left({k}+\mathrm{1}\right)^{\mathrm{2}} }=\frac{\mathrm{1}}{{k}^{\mathrm{2}} }−\frac{\mathrm{1}}{\left({k}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\Sigma=\left(\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{2}} }−\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }\right)+\left(\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{2}} }−\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2}} }\right)+...+\left(\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{2}} }−\frac{\mathrm{1}}{\mathrm{6}^{\mathrm{2}} }\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{2}} }−\frac{\mathrm{1}}{\mathrm{6}^{\mathrm{2}} }=\mathrm{1}−\frac{\mathrm{1}}{\mathrm{36}}=\frac{\mathrm{35}}{\mathrm{36}} \\ $$$$\frac{\mathrm{35}}{\mathrm{36}}{x}<\mathrm{70} \\ $$$${x}<\frac{\mathrm{70}×\mathrm{36}}{\mathrm{35}}=\mathrm{72} \\ $$$$\Rightarrow{x}_{{max}} =\mathrm{71}\:\Rightarrow{answer}\:{b} \\ $$

Commented by HongKing last updated on 29/Oct/21

alot thankyou ser cool

$$\mathrm{alot}\:\mathrm{thankyou}\:\mathrm{ser}\:\mathrm{cool} \\ $$

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