SO SÁNH:
A = \(\frac{2017^{2017}}{1+2017+2017^2+...+2017^{2016}}\)
B = \(\frac{2016^{2017}}{1+2016+2016^2+...+2016^{2016}}\)
So sánh A=\(\frac{2017^{2017}}{1+2017+2017^2+....+2017^{2016}}\)
B=\(\frac{2016^{2017}}{1+2016+2016^2+...+2016^{2016}}\)
Đặt C = 1 + 2017 + 20172 + ... + 20172016 ; D = 1 + 2016 + 20162 + ... + 20162016
Ta có : 2017C = 2017 + 20172 + 20173 + ... + 20172017
=> 2016C = 2017C - C = 20172017 - 1\(\Rightarrow C=\frac{2017^{2017}-1}{2016}\)
2016D = 2016 + 20162 + 20163 + ... + 20162017
=> 2015D = 2016D - D = 20162017 - 1\(\Rightarrow D=\frac{2016^{2017}-1}{2015}\)
\(\Rightarrow A=\frac{2017^{2017}}{\frac{2017^{2017}-1}{2016}}=\frac{2017^{2017}.2016}{2017^{2017}-1}\);\(B=\frac{2016^{2017}}{\frac{2016^{2017}-1}{2015}}=\frac{2016^{2017}.2015}{2016^{2017}-1}\)
Ta có : 20172017.2016.(20162017 - 1) - 20162017.2015.(20172017 - 1)
= 20172017.20162017.2016 - 20172017.2016 - 20172017.20162017.2015 + 20162017.2015
= 20172017.20162017 - 20172017.2016 + 20162017.2015
= 20172017.(20162017 - 2016) + 20162017.2015 > 0
=> A > B
Ta có
\(A=1:\frac{1+2017+2017^2+...+2017^{2016}}{2017^{2017}}\)
\(B=1:\frac{1+2016+2016^2+...2016^{2016}}{2016^{2017}}\)
\(A=1:\left(\frac{1}{2017^{2017}}+\frac{1}{2017^{2016}}+\frac{1}{2017^{2015}}+...+\frac{1}{2017}\right)\)
\(B=1:\left(\frac{1}{2016^{2017}}+\frac{1}{2016^{2016}}+\frac{1}{2016^{2015}}+...+\frac{1}{2016}\right)\)
Có 20172017>20162017 ; 20172016>20162016 ; 20172015>20162015;..... ; 2017>2016
=> \(\frac{1}{2017^{2017}}< \frac{1}{2016^{2017}};\frac{1}{2017^{2016}}< \frac{1}{2016^{2016}};\frac{1}{2017^{2015}}< \frac{1}{2016^{2015}};...;\frac{1}{2017}< \frac{1}{2016}\)
=> \(\frac{1}{2017^{2017}}+\frac{1}{2017^{2016}}+\frac{1}{2017^{2015}}+...+\frac{1}{2017}< \frac{1}{2016^{2017}}+\frac{1}{2016^{2016}}+\frac{1}{2016^{2015}}+...+\frac{1}{2016}\)
=> A>B ( vì số bị chia và số chia của A và B đều dương, số bị chia của cả 2 đều là 1, cái nào có số chia nhỏ hơn thì lớn hơn)
Xét biểu thức \(N=1+k+k^2+k^3+...+k^n\) (1) với k là số tự nhiên lớn hơn 1
Ta có \(k.N=k+k^2+k^3+k^4+...+k^{n+1}\) (2)
Lấy (2) - (1) ta được:
\(\left(k-1\right)N=\left(k+k^2+k^3+k^4+...+k^{n+1}\right)-\left(1+k+k^2+k^3+...+k^n\right)=k^{n+1}-1\)
Suy ra \(N=\frac{k^{n+1}-1}{k-1}\)
Áp dụng với k = 2017; n = 2016 ta được \(1+2017+2017^2+...+2017^{2016}=\frac{2017^{2017}-1}{2016}\)
Áp dụng với k = 2016; n = 2016 ta được \(1+2016+2016^2+...+2016^{2016}=\frac{2016^{2017}-1}{2015}\)
\(A=\frac{2017^{2017}}{1+2017+2017^2+...+2017^{2016}}=\frac{2017^{2017}}{\frac{2017^{2017}-1}{2016}}=\frac{2016.2017^{2017}}{2017^{2017}-1}>1\)
Tương tự \(B=\frac{2015.2016^{2017}}{2016^{2017}-1}>1\)
Mặt khác: Tử số A > tử số B; mẫu A > mẫu B => A < B.
So sánh A= \(\frac{2017^{2017}}{1+2017+2017^2+...+2017^{2016}}\)
B= \(\frac{2016^{2017}}{1+2016+2016^2+...+2016^{2016}}\)
SO SÁNH A VÀ B:
A = \(\frac{2017^{2016}}{1+2017+2017^2+...+2017^{2016}}\)
B = \(\frac{2016^{2017}}{1+2016+2016^2+...+2016^{2016}}\)
so sánh A và B, biết:
A= \(\frac{2017^{2017}}{1+2017+2017^2+...+2017^{2016}}\) B= \(\frac{2016^{2017}}{1+2016+2016^2+...+2016^{2016}}\)
Câu1: So sánh:
A = \(\frac{2016^{2017}+2}{2016^{2017}-1}\) và B = \(\frac{2016^{2017}}{2016^{2017}-3}\)
Vì B > 1
=> \(B=\frac{2016^{2017}}{2016^{2017}-3}>\frac{2016^{2017}+2}{2016^{2017}-3+2}=\frac{2016^{2017}+2}{2016^{2017}-1}=A\)
Vậy A=B
so sanh A va B
\(A=\frac{2017^{100}}{1+2017+2017^2+2017^3+...+2017^{100}}\)
\(B=\frac{2016^{100}}{1+2016+2016^2+2016^3+...+2016^{100}}\)
Ta có: \(A=\frac{2017^{100}}{1+2017+2017^2+2017^3+...+2017^{100}}\)
\(\Leftrightarrow A=\frac{\left[\left(20.100\right)+16+1\right]^{100}}{1+2017+2017^2+2017^3+...+2017^{10}}\)
\(B=\frac{2016^{100}}{1+2016+2016^2+2016^3+...+2016^{100}}\)
\(\Leftrightarrow B=\frac{\left[\left(20.100+16\right)\right]^{100}}{1+2016+2016^2+2016^3+...+2016^{100}}\)
Ta có hai tổng A và B mới để so sánh:
\(A=\frac{\left[\left(20.100\right)+16+1\right]^{100}}{1+2017+2017^2+2017^3+...+2017^{100}}\)
\(B=\frac{\left[\left(20.100\right)+16\right]^{100}}{1+2016+2016^2+2016^3+...+2016^{100}}\)
Tới đây đơn giản rồi. Bạn làm tiếp đi nhé! Mẹ mình bắt tắt máy không cho làm nên đành dừng lại ở đây thôi! Thông cảm :V
So sánh 2 phân số
A=\(\frac{2015}{2016}+\frac{2016}{2017}+\frac{2017}{2018}\)
B=\(\frac{2015+2016+2017}{2016+2017+2018}\)
Ta có : \(B=\frac{2015+2016+2017}{2016+2017+2018}\) \(=\frac{2015}{2016+2017+2018}+\frac{2016}{2016+2017+2018}+\frac{2017}{2016+2017+2018}\)
Mà \(\frac{2015}{2016}>\frac{2015}{2016+2017+2018}\)
\(\frac{2016}{2017}>\frac{2016}{2016+2017+2018}\)
\(\frac{2017}{2018}>\frac{2017}{2016+2017+2016}\)
Cộng vế theo vế, ta có :
\(\frac{2015}{2016}+\frac{2016}{2017}+\frac{2017}{2018}>\frac{2015+2016+2017}{2016+2017+2018}\)
\(\Rightarrow A>B\)
So sánh hai phân số : A=\(\frac{2015}{2016}+\frac{2016}{2017}+\frac{2017}{2018}\)và B=\(\frac{2015+2016+2017}{2016+2017+2018}\)
\(A=\frac{2015}{2016}+\frac{2016}{2017}+\frac{2017}{2018}\)
\(B=\frac{2015+2016+2017}{2016+2017+2018}\)
\(B=\frac{2015}{2016+2017+2018}+\frac{2016}{2016+2017+2018}+\frac{2017}{2016+2017+2018}\)
Ta có:
\(\frac{2015}{2016}>\frac{2015}{2016+2017+2018}\)
\(\frac{2016}{2017}>\frac{2016}{2016+2017+2018}\)
\(\frac{2017}{2018}>\frac{2017}{2016+2017+2018}\)
Cộng vế theo vế, ta có:
\(\frac{2015}{2016}+\frac{2016}{2017}+\frac{2017}{2018}>\frac{2015}{2016+2017+2018}+\frac{2016}{2016+2017+2018}+\frac{2017}{2016+2017+2018}\)
\(hay\frac{2015}{2016}+\frac{2016}{2017}+\frac{2017}{2018}>\frac{2015+2016+2017}{2016+2017+2018}\)
\(\Rightarrow A>B\)
Vậy A > B
So sánh
\(A=\frac{2017^{2016}-2020}{2017^{2016}-6056}\)
\(B=\frac{2017^{2016}+2018}{2017^{2016}-2018}\)