## $S=$ $\frac{1}{4}$ + $\frac{1}{4^{2}}$ + $\frac{1}{4^{3}}$ + … + $\frac{1}{4^{50}}$

Question

$S=$ $\frac{1}{4}$ + $\frac{1}{4^{2}}$ + $\frac{1}{4^{3}}$ + … + $\frac{1}{4^{50}}$

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3 tuần 2021-11-21T18:58:33+00:00 2 Answers 4 views 0

1. S=1/4+1/4^2+1/4^3+…+1/4^50

=> 1/4 S = 1/4(1/4+1/4^2+1/4^3+…+1/4^50)

⇒ 1/4 S = 1/4^2 + 1/4^3 + 1/4^4 +…+ 1/4^51

⇒ S – 1/4 S = (1/4+1/4^2+1/4^3+…+1/4^50)-(1/4^2 + 1/4^3 + 1/4^4 +…+ 1/4^51)

⇒ 3/4 S = 1/4 – 1/4^51

⇒ S = (1/4 – 1/4^51) : 3/4

⇒ S = (1/4 – 1/4^51) . 4/3

⇒ S = 1/3 – 1/(4^50 . 3)

2. $S=\frac{1}{4}+\frac{1}{4^2}+\frac{1}{4^3}+…+\frac{1}{4^{50}}$

$⇒4S=\frac{4}{4}+\frac{4}{4^2}+\frac{4}{4^3}+…+\frac{4}{4^{50}}$

$⇒4S=1+\frac{1}{4}+\frac{1}{4^2}+…+\frac{1}{4^{49}}$

$⇒4S-S=(1+\frac{1}{4}+\frac{1}{4^2}+…+\frac{1}{4^{49}})-(\frac{1}{4}+\frac{1}{4^2}+\frac{1}{4^3}+…+\frac{1}{4^{50}})$

$⇒3S=1-\frac{1}{4^{50}}$

$⇒S=\frac{1}{3}-\frac{1}{3.4^{50}}$.