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Question Number 186468 by aba last updated on 04/Feb/23

Simplify ((1^2 ∙2!+2^2 ∙3!+3^2 ∙4!+∙∙∙+n^2 (n+1)!−2)/((n+1)!)) to n^2 +n−2

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{Simplify} \\ $$$$\frac{\mathrm{1}^{\mathrm{2}} \centerdot\mathrm{2}!+\mathrm{2}^{\mathrm{2}} \centerdot\mathrm{3}!+\mathrm{3}^{\mathrm{2}} \centerdot\mathrm{4}!+\centerdot\centerdot\centerdot+{n}^{\mathrm{2}} \left({n}+\mathrm{1}\right)!−\mathrm{2}}{\left({n}+\mathrm{1}\right)!} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{to} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{n}^{\mathrm{2}} +\mathrm{n}−\mathrm{2} \\ $$

Answered by SEKRET last updated on 07/Feb/23

1^2 ∙2=2 2^2 ∙3!=4! 3^2 ∙4!=2∙5!−4! 4^2 ∙5!=3∙6!−2∙5! 5^2 ∙6!=4∙7!-3∙6! ....... ............ n^2 (n+1)!=(n−1)∙(n+2)! −(n−2)∙(n+1)! (+) all 2+(n−1)∙(n+2)! ((2+(n−1)∙(n+2)! −2)/((n+1)!))=(((n−1)∙(n+2)(n+1)!)/((n+1)!)) = (n−1)(n+2)=n^2 +n−2 ABDULAZIZ ABDUVALIYEV

$$\:\:\:\:\:\mathrm{1}^{\mathrm{2}} \centerdot\mathrm{2}=\mathrm{2} \\ $$$$\:\:\:\:\:\mathrm{2}^{\mathrm{2}} \centerdot\mathrm{3}!=\mathrm{4}! \\ $$$$\:\:\:\:\:\mathrm{3}^{\mathrm{2}} \centerdot\mathrm{4}!=\mathrm{2}\centerdot\mathrm{5}!−\mathrm{4}! \\ $$$$\:\:\:\:\mathrm{4}^{\mathrm{2}} \centerdot\mathrm{5}!=\mathrm{3}\centerdot\mathrm{6}!−\mathrm{2}\centerdot\mathrm{5}! \\ $$$$\:\:\:\:\mathrm{5}^{\mathrm{2}} \centerdot\mathrm{6}!=\mathrm{4}\centerdot\mathrm{7}!-\mathrm{3}\centerdot\mathrm{6}! \\ $$$$\:\:....... \\ $$$$............ \\ $$$$\:\:\:\boldsymbol{\mathrm{n}}^{\mathrm{2}} \left(\boldsymbol{\mathrm{n}}+\mathrm{1}\right)!=\left(\boldsymbol{\mathrm{n}}−\mathrm{1}\right)\centerdot\left(\boldsymbol{\mathrm{n}}+\mathrm{2}\right)!\:−\left(\boldsymbol{\mathrm{n}}−\mathrm{2}\right)\centerdot\left(\boldsymbol{\mathrm{n}}+\mathrm{1}\right)! \\ $$$$\left(+\right)\:\:\boldsymbol{\mathrm{all}} \\ $$$$\:\:\:\:\mathrm{2}+\left(\boldsymbol{\mathrm{n}}−\mathrm{1}\right)\centerdot\left(\boldsymbol{\mathrm{n}}+\mathrm{2}\right)! \\ $$$$\:\frac{\mathrm{2}+\left(\boldsymbol{\mathrm{n}}−\mathrm{1}\right)\centerdot\left(\boldsymbol{\mathrm{n}}+\mathrm{2}\right)!\:−\mathrm{2}}{\left(\boldsymbol{\mathrm{n}}+\mathrm{1}\right)!}=\frac{\left(\boldsymbol{\mathrm{n}}−\mathrm{1}\right)\centerdot\left(\boldsymbol{\mathrm{n}}+\mathrm{2}\right)\left(\boldsymbol{\mathrm{n}}+\mathrm{1}\right)!}{\left(\boldsymbol{\mathrm{n}}+\mathrm{1}\right)!} \\ $$$$=\:\left(\boldsymbol{\mathrm{n}}−\mathrm{1}\right)\left(\boldsymbol{\mathrm{n}}+\mathrm{2}\right)=\boldsymbol{\mathrm{n}}^{\mathrm{2}} +\boldsymbol{\mathrm{n}}−\mathrm{2} \\ $$$$\:\:\boldsymbol{{ABDULAZIZ}}\:\:\boldsymbol{{ABDUVALIYEV}} \\ $$$$\:\: \\ $$

Commented by aba last updated on 11/Feb/23

thank a lot��

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