ngu

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.

A = \(\frac{\frac{3}{4}-\frac{3}{11}+\frac{3}{13}}{\frac{5}{4}-\frac{5}{11}+\frac{5}{13}}+\frac{\frac{1}{2}-\frac{1}{3}+\frac{1}{4}}{\frac{5}{4}-\frac{5}{6}+\frac{5}{8}}\)

\(=\frac{3.\left(\frac{1}{4}-\frac{1}{11}+\frac{1}{13}\right)}{5.\left(\frac{1}{4}-\frac{1}{11}+\frac{1}{13}\right)}+\frac{\frac{1}{2}-\frac{1}{3}+\frac{1}{4}}{\frac{5}{2}.\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{4}\right)}\)

\(=\frac{3}{5}+\frac{1}{\frac{5}{2}}\)

\(=\frac{3}{5}+\frac{2}{5}=1\)

b) B = \(\frac{2^{12}.3^5-4^6.9^2}{\left(2^2.3\right)^6.8^4.3^5}-\frac{5^{10}.7^3:25^5.49}{\left(125.7\right)^3+5^9.14^3}\)

\(=\frac{2^{12}.3^5-\left(2^2\right)^6.\left(3^2\right)^2}{2^{12}.3^6+\left(2^3\right)^4.3^5}-\frac{5^{10}.7^3-\left(5^2\right)^5.7^2}{\left(5^3\right)^3.7^3+5^9.\left(7.2\right)^3}\)

\(=\frac{2^{12}.3^5-2^{12}.3^4}{2^{12}.3^6+2^{12}.3^5}-\frac{5^{10}.7^3-5^{10}-7^2}{5^9.7^3+5^9.7^3.2^3}\)

\(=\frac{2^{12}.3^4.\left(3-1\right)}{2^{12}.3^5\left(3+1\right)}-\frac{5^{10}.7^2.\left(7-1\right)}{5^9.7^3\left(1+2^3\right)}\)

 \(=\frac{1}{3.2}-\frac{5.2}{7.3}\)

\(=\frac{7}{3.2.7}-\frac{5.2.2}{7.3.2}\)

\(=\frac{7}{42}-\frac{20}{42}\)

\(=-\frac{13}{42}\)

cs ng làm đung r

đag định lm

Bằng \(-\frac{2}{11}\)