Cho S = 1+2+2^2+2^3+ … + 2^9 Hãy so sánh S với 5 . 2^8

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1. Ta có:

   ` S = 1 + 2 + 2^2 + 2^3 + … + 2^9 `

   ` <=> 2S = 2 + 2^2 + 2^3 + 2^4 + … + 2^{10} `

   ` <=> 2S – S = 2 + 2^2 + 2^3 + 2^4 + … + 2^{10} – 1 – 2 – 2^2 – 2^3 – … – 2^9 `

   ` <=> S = (2 – 2) + (2^2 – 2^2) + … + (2^9 – 2^9) + 2^{10} – 1 `

   ` <=> S = 2^{10} – 1 `

   Lại có:

   ` 5 . 2^8 `

   ` = (4 + 1) . 2^8 `

   ` = 2^8 . 4 + 2^8 `

   ` = 2^{10} + 2^{8} `

   Vì ` 2^8 > -1 `

   ` => 2^{10} + 2^{8} > + 2^{10} – 1 `

   ` => 5 . 2^{8} > S ` 

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2. $S=1+2+2^2+…+2^9$

   $2S=2+2^2+2^3+…+2^{10}$

   $2S-S=(2+2^2+2^3+…+2^{10})-(1+2+2^2+…+2^9)$

   $S=2^{10}-1$

   $S=2^8.4-1$

   Ta thấy: $5.2^8>2^8.4$

   $→5.2^8>2^8.4-1$

   $→5.2^8>S$

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