Toán Cho S = 1+2+2^2+2^3+ … + 2^9 Hãy so sánh S với 5 . 2^8 10/07/2021 By Charlie Cho S = 1+2+2^2+2^3+ … + 2^9 Hãy so sánh S với 5 . 2^8
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 ` Trả lời
$S=1+2+2^2+…+2^9$ $2S=2+2^2+2^3+…+2^{10}$ $2S-S=(2+2^2+2^3+…+2^{10})-(1+2+2^2+…+2^9)$ $S=2^{10}-1$ $S=2^8.4-1$ Ta thấy: $5.2^8>2^8.4$ $→5.2^8>2^8.4-1$ $→5.2^8>S$ Trả lời
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 `
$S=1+2+2^2+…+2^9$
$2S=2+2^2+2^3+…+2^{10}$
$2S-S=(2+2^2+2^3+…+2^{10})-(1+2+2^2+…+2^9)$
$S=2^{10}-1$
$S=2^8.4-1$
Ta thấy: $5.2^8>2^8.4$
$→5.2^8>2^8.4-1$
$→5.2^8>S$