Tìm $x$, biết : $\ \dfrac{2020^{100} + 2020^{96} + 2020^{92} + … + 2020^{4} + 1}{|x – 2020|} = \dfrac{2020^{104} – 1}{2020^{4} – 1}$

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1. Tham khảo

    Do đó `(2020^{100}+2020^{96}+…+2020^4+1).(2020^4-1)=|x-2020|.(2020^{104}-1)`

   Đặt `A= 2020^{100}+2020^{96}+…+2020^4+1`

   `⇒2020^{4}A=2020^{104}+2020^{100}+…+2020^{8}+2020^4`

   `⇒2020^{4}A-A=2020^{104}+2020^{100}+…+2020^{8}+2020^4-(2020^{100}+2020^{96}+…+2020^4+1)`

   `⇒(2020^{4}-1)A=2020^{104}-1`

   `⇒A=\frac{2020^{104}-1}{2020^{4}-1}`

   Nên `\frac{2020^{104}-1}{2020^{4}-1}×(2020^{4}-1)=|x-2020|.(2020^{104}-1)`

   `⇒2020^{104}-1=|x-2020|.(2020^{104}-1)`

   `⇒|x-2020|=1`

   `⇒`\(\left[ \begin{array}{l}x-2020=1\\x-2020=-1\end{array} \right.\) 

   `⇒`\(\left[ \begin{array}{l}x=2020+1\\x=-1+2020\end{array} \right.\) 

   `⇒`\(\left[ \begin{array}{l}x=2021\\x=2019\end{array} \right.\)

   Vậy `x=2021` hoặc `x=2019`

   `\text{©CBT}`

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