Ta có:

M=\(\frac{1}{2}.\frac{3}{4}.....\frac{99}{100}\)

M=\(\frac{1.3....99}{2.4....100}\)

Lại có:

N=\(\frac{2}{3}.\frac{4}{5}....\frac{100}{101}\)

N=\(\frac{2.4....100}{3.5....101}\)

\(\Rightarrow\)M.N=\(\frac{1.2.3......99.100}{2.3.4......100.101}\)

\(\Rightarrow\)M.N=\(\frac{1}{101}\)

1.

Ta có:

1/2 < 2/3

3/4 < 4/5

.............

99/100 < 100/101

=> 1/2*3/4*5/6*...*99/100 < 2/3*4/5*6/7*...*100/101

=> A < B

2.

\(A\cdot B=\left[\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdot...\cdot\frac{99}{100}\right]\cdot\left[\frac{2}{3}\cdot\frac{4}{5}\cdot\frac{6}{7}\cdot...\cdot\frac{100}{101}\right]\)

\(A\cdot B=\frac{\left[1\cdot3\cdot5\cdot7\cdot...\cdot99\right]\left[2\cdot4\cdot6\cdot8\cdot...\cdot100\right]}{\left[2\cdot4\cdot6\cdot8\cdot...\cdot100\right]\left[3\cdot5\cdot7\cdot9\cdot...\cdot101\right]}=\frac{1\cdot3\cdot5\cdot...\cdot99}{3\cdot5\cdot7\cdot...\cdot101}=\frac{1}{101}\)

3.

Vì A < B => A.A < A.B => A² < 1/101 < 1/100

Mà A² < 1/100 <=> A² < \(\frac{1}{10}^2\)=> A < 1/10

\(A=3+3^2+3^3+...+3^{100}\)

\(\Leftrightarrow3A=3^2+3^3+3^4+3^5+....+3^{101}\)

\(\Leftrightarrow3A-A=\left(3^2+3^3+3^4+3^5+...+3^{101}\right)-\left(3+3^2+3^3+3^4+...+3^{100}\right)\)

\(\Leftrightarrow2A=3^{101}-3\)

\(\Leftrightarrow A=\frac{3^{101}-3}{2}< 3^{100}-1\)

\(\Leftrightarrow A< B\)

a. tính A = 3+3^2+3^3+3^4+.....+3^100

3A=3^2+3^3+3^4+3^5+....+3^100

3A-A=(3^2+3^3+3^4+....+3^101)-(3+3^2+3^3+3^4+.....+3^100)=3^101-3=3^100

mà B=3^100-1 => A<B

\(A=1+4+4^2+...+4^{99}\)

\(\Leftrightarrow4A=4+4^2+4^3+...+4^{100}\)

\(\Leftrightarrow3A=4^{100}-1\)

\(\Leftrightarrow A=\frac{4^{100}-1}{3}< \frac{4^{100}}{3}\)

hay A<B (đpcm)