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\(n=2^{2019}-2^{2018}-...-2^1-1=2^{2019}-\left(2^{2018}+2^{2017}+...+2^1+1\right)\)
Đặt\(S=1+2+...+2^{2017}+2^{2018}\)
\(\Rightarrow2S=2+2^2+...+2^{2018}+2^{2019}\)
\(\Rightarrow2S-S=\left(2+2^2+...+2^{2018}+2^{2019}\right)-\left(1+2+...+2^{2017}+2^{2018}\right)\)
\(\Rightarrow S=2^{2019}-1\)
Mà\(n=2^{2019}-S\)
\(\Rightarrow n=2^{2019}-\left(2^{2019}-1\right)=1\)
\(\Rightarrow A=3^1+2^1+2020^1=2025\)
Happy new year :)))
Ta có : n = 22019 - 22018 - 22017 - .... - 22 - 2 - 1 (1)
=> 2n = 22020 - 22019 - 22018 - .... - 23 - 22 - 2 (2)
Lấy (2) trừ (1) theo vế ta có :
2n - n = (22020 - 22019 - 22018 - .... - 23 - 22 - 2) - (22019 - 22018 - 22017 - .... - 22 - 2 - 1)
=> n = 22020 - 22019 - 22019 + 1
=> n = 22020 - 2.22019 + 1 = 22020 - 22020 + 1 = 1
Khi đó A = 31 + 21 + 20201 = 3 + 2 + 2020 = 2025
Vậy A = 2025
\(M=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2017}-\frac{1}{2018}+\frac{1}{2019}\)
\(=\left(1+\frac{1}{3}+...+\frac{1}{2017}+\frac{1}{2019}\right)-\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2018}\right)\)
\(=\left(1+\frac{1}{2}+...+\frac{1}{2018}+\frac{1}{2019}\right)-2\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2018}\right)\)
\(=\left(1+\frac{1}{2}+...+\frac{1}{2018}+\frac{1}{2019}\right)-\left(1+\frac{1}{2}+...+\frac{1}{1009}\right)\)
\(=\frac{1}{1010}+\frac{1}{1011}+...+\frac{1}{2019}=N\)
\(\Rightarrow M-N=0\Rightarrow\left(M-N\right)^2=0\)
\(\left(|x|-2017\right)^{\left(n+2018\right)\cdot\left(n+2019\right)}=-\left(2^3-3^2\right)^{2019}\)
\(\left(\left|x\right|-2017\right)^{\left(n+2018\right)\left(n+2019\right)}=-\left(2^3-3^2\right)^{2019}\)
\(\left(\left|x\right|-2017\right)^{\left(n+2018\right)\left(n+2019\right)}=-\left(-1\right)^{2019}=1\)
\(\Rightarrow\orbr{\begin{cases}\left(n+2018\right)\left(n+2019\right)=0\\\left|x\right|-2017=1\end{cases}}\)
\(\Rightarrow\orbr{\begin{cases}\orbr{\begin{cases}n=-2018\\n=-2019\end{cases}}\\\orbr{\begin{cases}x=2018\\x=-2018\end{cases}}\end{cases}}\)
Theo bài ra, ta có: \(B=\dfrac{2018}{1}+\dfrac{2017}{2}+\dfrac{2016}{3}+...+\dfrac{1}{2018}\)
\(B=\left(\dfrac{2018}{1}+1\right)+\left(\dfrac{2017}{2}+1\right)+\left(\dfrac{2016}{3}+1\right)+...+\left(\dfrac{1}{2018}+1\right)-2018\)
\(B=2019+\dfrac{2019}{2}+\dfrac{2019}{3}+...+\dfrac{2019}{2018}-2018\)
\(B=\dfrac{2019}{2}+\dfrac{2019}{3}+...+\dfrac{2019}{2018}+\left(2019-2018\right)\)
\(B=\dfrac{2019}{2}+\dfrac{2019}{3}+...+\dfrac{2019}{2018}+1\)
\(B=\dfrac{2019}{2}+\dfrac{2019}{3}+...+\dfrac{2019}{2018}+\dfrac{2019}{2019}\)
\(B=2019\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2019}\right)\)
Khi đó:\(\dfrac{B}{A}=\dfrac{2019\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2019}\right)}{\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2019}}\)
\(\Rightarrow\dfrac{B}{A}=2019\), là 1 số nguyên.
Vậy \(\dfrac{B}{A}\) là số nguyên.
\(\frac{A}{B}=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2020}}{\frac{1}{2019}+\frac{2}{2018}+\frac{3}{2017}+...+\frac{2018}{2}+\frac{2019}{1}}\)
\(\frac{A}{B}=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2020}}{\frac{1}{2019}+1+\frac{2}{2018}+1+\frac{3}{2017}+1+...+\frac{2018}{2}+1+1}\)
\(\frac{A}{B}=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2020}}{\frac{2020}{2019}+\frac{2020}{2018}+\frac{2020}{2017}+...+\frac{2020}{2}+\frac{2020}{2020}}\)
\(\frac{A}{B}=\frac{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2020}}{2020\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2020}\right)}\)
\(\frac{A}{B}=\frac{1}{2020}\)
1
\(A=\frac{2019^{2019}+1}{2019^{2020}+1}< \frac{2019^{2019}+1+2018}{2019^{2020}+1+2018}=\frac{2019^{2019}+2019}{2019^{2020}+2019}=\frac{2019\left(2019^{2018}+1\right)}{2019\left(2019^{2019}+1\right)}\)
\(=\frac{2019^{2018}+1}{2019^{2019}+1}\)
2
\(M=\frac{100^{101}+1}{100^{100}+1}< \frac{100^{101}+1+99}{100^{100}+1+99}=\frac{100^{101}+100}{100^{100}+100}=\frac{100\left(100^{100}+1\right)}{100\left(100^{99}+1\right)}\)
\(=\frac{100^{100}+1}{100^{99}+1}=N\)
3.
\(2^x=256+2^y\\ \Rightarrow2^x-2^y=256\\ \Rightarrow2^y\left(2^{x-y}-1\right)=2^8\)
\(\Rightarrow2^y;2^{x-y}-1\in U\left(2^8\right)\)
Mà \(2^{x-y}-1\) là số lẻ
\(\Rightarrow2^{x-y}-1=1\\ \Rightarrow\left\{{}\begin{matrix}2^y=2^8\\2^{x-y}=2\end{matrix}\right.\\ \Rightarrow\left\{{}\begin{matrix}y=8\\x-y=1\end{matrix}\right.\\ \Rightarrow\left\{{}\begin{matrix}y=8\\x=9\end{matrix}\right.\)
4.
Gọi d là ƯCLN(2n+5;3n+7)
\(\Rightarrow\left\{{}\begin{matrix}2n+5⋮d\\3n+7⋮d\end{matrix}\right.\\ \Rightarrow\left\{{}\begin{matrix}3\left(2n+5\right)⋮d\\2\left(3n+7\right)⋮d\end{matrix}\right.\\ \Rightarrow\left\{{}\begin{matrix}6n+15⋮d\\6n+14⋮d\end{matrix}\right.\\ \Rightarrow\left(6n+15\right)-\left(6n+14\right)⋮d\\ \Rightarrow1⋮d\\ \Rightarrow d=1\)
=> đpcm
Nguyễn Huy Tú lê thị hương giang Hồng Phúc Nguyễn
Nguyễn Thanh Hằng Akai Haruma Nam Nguyễn Hà Nam Phan Đình
Aki Tsuki
\(N=2^{2019}-2^{2018}-2^{2017}-...-2-1\)
\(=2^{2019}-\left(2^{2018}+2^{2017}+...+2+1\right)\)
Đặt \(B=1+2+...+2^{2017}+2^{2018}\)
\(\Rightarrow\) \(2B=2+2^2+...+2^{2018}+2^{2019}\)
\(\Rightarrow\) \(B=2^{2019}-1\)
\(\Rightarrow\) \(N=2^{2019}-2^{2019}+1=1\)
\(\Rightarrow\) \(A=20^1+11^1+2019^1\)
\(=20+11+2019\)
\(=2050\)
Study well ! >_<
N=\(2^{2019}-\left(1+2+.....2^{2018}\right)\)
Đặt B=\(1+2+..........+2^{2018}\)
2B=\(2+2^2+..........+2^{2019}\)
2B-B=B=\(2^{2019}-1\)
Suy ra N=\(2^{2019}-2^{2019}+1=1\)
A=20+11+2019=2050
hok tốt