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Question Number 153839 by liberty last updated on 11/Sep/21

Commented by mathdanisur last updated on 11/Sep/21

x(1 + (1/(1+2)) + (1/(1+2+3)) + ... + (1/(1+2+...+4041))) = 4041 ▲ A = 1 + (1/(1+2)) + (1/(1+2+3)) + ... + (1/(1+2+...+n)) = ((2n)/(n + 1)) n = 4041 ⇒ A = ((2 ∙ 4041)/(4041 + 1)) = ((2 ∙ 4041)/(2 ∙ 2021)) = ((4041)/(2021)) ⇒ x ∙ ((4041)/(2021)) = 4041 ⇒ x = 2021 ▲

$$\mathrm{x}\left(\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{1}+\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{1}+\mathrm{2}+\mathrm{3}}\:+\:...\:+\:\frac{\mathrm{1}}{\mathrm{1}+\mathrm{2}+...+\mathrm{4041}}\right)\:=\:\mathrm{4041} \\ $$$$\blacktriangle\:\boldsymbol{\mathrm{A}}\:=\:\mathrm{1}\:+\:\frac{\mathrm{1}}{\mathrm{1}+\mathrm{2}}\:+\:\frac{\mathrm{1}}{\mathrm{1}+\mathrm{2}+\mathrm{3}}\:+\:...\:+\:\frac{\mathrm{1}}{\mathrm{1}+\mathrm{2}+...+\boldsymbol{\mathrm{n}}}\:=\:\frac{\mathrm{2}\boldsymbol{\mathrm{n}}}{\boldsymbol{\mathrm{n}}\:+\:\mathrm{1}} \\ $$$$\boldsymbol{\mathrm{n}}\:=\:\mathrm{4041}\:\Rightarrow\:\boldsymbol{\mathrm{A}}\:=\:\frac{\mathrm{2}\:\centerdot\:\mathrm{4041}}{\mathrm{4041}\:+\:\mathrm{1}}\:=\:\frac{\mathrm{2}\:\centerdot\:\mathrm{4041}}{\mathrm{2}\:\centerdot\:\mathrm{2021}}\:=\:\frac{\mathrm{4041}}{\mathrm{2021}} \\ $$$$\Rightarrow\:\boldsymbol{\mathrm{x}}\:\centerdot\:\frac{\mathrm{4041}}{\mathrm{2021}}\:=\:\mathrm{4041}\:\Rightarrow\:\boldsymbol{\mathrm{x}}\:=\:\mathrm{2021}\:\blacktriangle \\ $$

Answered by puissant last updated on 11/Sep/21

1+(1/(1+2))+(1/(1+2+3))+....+(1/(1+2+3+...+4041)) =(1/((4042)/(2×4041))) = ((2×4041)/(4042)).. ⇒ x(1+(1/(1+2))+....+(1/(1+2+..+4041)))=4041 ⇒ (((2×4041)x)/(4042))=4041 ⇒ ((2x)/(4042))=1 ⇒ 2x=4042 ⇒ x=2021.. ∴∵ x=2021....■

$$\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{2}+\mathrm{3}}+....+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{2}+\mathrm{3}+...+\mathrm{4041}} \\ $$$$=\frac{\mathrm{1}}{\frac{\mathrm{4042}}{\mathrm{2}×\mathrm{4041}}}\:=\:\frac{\mathrm{2}×\mathrm{4041}}{\mathrm{4042}}.. \\ $$$$\Rightarrow\:{x}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{2}}+....+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{2}+..+\mathrm{4041}}\right)=\mathrm{4041} \\ $$$$\Rightarrow\:\frac{\left(\mathrm{2}×\mathrm{4041}\right){x}}{\mathrm{4042}}=\mathrm{4041} \\ $$$$\Rightarrow\:\frac{\mathrm{2}{x}}{\mathrm{4042}}=\mathrm{1}\:\Rightarrow\:\mathrm{2}{x}=\mathrm{4042}\:\Rightarrow\:{x}=\mathrm{2021}.. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\therefore\because\:\:\:\:{x}=\mathrm{2021}....\blacksquare \\ $$

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