Question

So sánh:

a,

\(\frac{10^{109}+2}{10^{110}-5}\)với \(\frac{10^{110}+2}{10^{111}-5}\)

b,

\(\frac{9^{65}-1}{9^{64}+2}\)với \(\frac{9^{70}-1}{9^{69}+2}\)

c,

\(\frac{1+11+11^2+....+11^{10}}{1+11+11^2+...+11^{11}}\)với \(\frac{1+13+13^2+..+13^{10}}{1+13+13^2+...+13^{11}}\)

d,

\(1+2+2^2+...+2^{80}\)với \(2^{81}-2\)

e,

\(\frac{3}{4}+\frac{8}{9}+\frac{15}{16}+..+\frac{899}{900}\) với 29

Answer 1

d)

đặt A = 1 + 2 + 2² + ... + 2⁸⁰ 

2A = 2 + 2² + 2³ + ... + 2⁸¹

2A - A = ( 2 + 2² + 2³ + ... + 2⁸¹ ) - ( 1 + 2 + 2² + ... + 2⁸⁰ )

A = 2⁸¹ - 1 > 2⁸¹ - 2

e) 

đặt \(A=\frac{3}{4}+\frac{8}{9}+\frac{15}{16}+...+\frac{899}{900}\)

\(A=\left(1-\frac{1}{4}\right)+\left(1-\frac{1}{9}\right)+\left(1-\frac{1}{16}\right)+...+\left(1-\frac{1}{900}\right)\)

\(A=\left(1+1+1+...+1\right)-\left(\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+...+\frac{1}{900}\right)\)

\(A=29-\left(\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+...+\frac{1}{900}\right)\)

đặt \(B=\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+...+\frac{1}{900}\)

\(B=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{30^2}\)

\(B< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{29.30}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{29}-\frac{1}{30}\)

\(=1-\frac{1}{30}=\frac{29}{30}< 1\)

\(\Rightarrow A< 29\)

Answer 2

So sánh C và D biết
C=1+13+13^2+...+13^13/1+13+13^2+...+13^12
D=1+11+11^2+...+11^13/1+11+11^2+...+11^12