Question

[Ngọc Diễm]

Bài 1: Chứng minh rằng tổng

S= \(\dfrac{1}{2^2}\)- \(\dfrac{1}{2^4}\)+\(\dfrac{1}{2^6}\)- ...+ \(\dfrac{1}{2^{4n-2}}\)-\(\dfrac{1}{2^{4n}}\)+... + \(\dfrac{1}{2^{2018}}\)-\(\dfrac{1}{2^{2020}}\) <0,2

Bài 2: Cho T= 2²⁰²⁰- 2²⁰¹⁹-2²⁰¹⁸-...-2-1. Tính 2021^T

Bài 3: Tìm số tự nhiên x, biết

\(\dfrac{7^{x+2}+7^{x+1}+7^x}{57}\)=\(\dfrac{5^{2x}+5^{2x+1}+5^{2x+3}}{131}\)

Bài 4: Tìm x, biết

a) \(\dfrac{1}{4}\). \(\dfrac{2}{6}\). \(\dfrac{3}{8}\)... \(\dfrac{30}{62}\). \(\dfrac{31}{64}\)= 4^x

b) \(\dfrac{4^5+4^5+4^5+4^5}{3^5+3^5+3^5}\). \(\dfrac{6^5+6^5+6^5+6^5+6^5+6^5}{2^5+2^5}\)= 8^x

     [ Giúp mình giải nhanh 1 trong 4 bài với ak ]

     [ Cảm ơn mọi người nha ] 

Answer 1

[Ngô Lê Kim Nhi]

Làm seo để viết phân số thế?

Answer 2

[HT.Phong (9A5)]

Bài 4:

a) 

\(\dfrac{1}{4}\cdot\dfrac{2}{6}\cdot\dfrac{3}{8}\cdot...\cdot\dfrac{30}{62}\cdot\dfrac{31}{64}=4^x\\ \Rightarrow\dfrac{1\cdot2\cdot3\cdot...\cdot30\cdot31}{4\cdot6\cdot8\cdot...\cdot62\cdot64}=4^x\\ \Rightarrow\dfrac{2\cdot3\cdot4\cdot...\cdot30\cdot31}{\left(2\cdot2\right)\cdot\left(3\cdot2\right)\cdot\left(4\cdot2\right)\cdot...\cdot\left(2\cdot31\right)\cdot\left(2\cdot32\right)}=\left(2^2\right)^x\\ \Rightarrow\dfrac{2\cdot3\cdot4\cdot...\cdot30\cdot31}{2^{31}\cdot\left(2\cdot3\cdot4\cdot...\cdot31\cdot32\right)}=2^{2x}\\ \Rightarrow\dfrac{1}{2^{31}\cdot32}=2^{2x}\\ \Rightarrow\dfrac{1}{2^{31}\cdot2^5}=2^{2x}\\ \Rightarrow\dfrac{1}{2^{36}}=2^{2x}\\ \Rightarrow2^{-36}=2^{2x}\\ \Rightarrow2x=-36\\ \Rightarrow x=-18\)

b) 

\(\dfrac{4^5+4^5+4^5+4^5}{3^5+3^5+3^5}\cdot\dfrac{6^5+6^5+6^5+6^5+6^5+6^5}{2^5+2^5}=8^x\\ \Rightarrow\dfrac{4\cdot4^5}{3\cdot3^5}\cdot\dfrac{6\cdot6^5}{2\cdot2^5}=8^x\\ \Rightarrow\dfrac{4^6}{3^6}\cdot\dfrac{6^6}{2^6}=8^x\\ \Rightarrow\dfrac{4^6}{3^6}\cdot3^6=8^x\\ \Rightarrow4^6=8^x\\ \Rightarrow\left(2^2\right)^6=\left(2^3\right)^x\\ \Rightarrow2^{12}=2^{3x}\\ \Rightarrow3x=12\\ \Rightarrow x=4\)