Question

Answer 1

a: 1/31<1/30

1/32<1/30

...

1/40<1/30

1/41<1/40

1/42<1/40

...

1/149<1/140

1/150<1/140

=>A=1/31+1/32+...+1/150<1/3+1/4+...+1/14=1,75<13/6

b: \(B=\dfrac{1}{2}\cdot\dfrac{3}{4}\cdot\dfrac{5}{6}\cdot...\cdot\dfrac{99}{100}\)

=>\(B< \dfrac{2}{3}\cdot\dfrac{4}{5}\cdot\dfrac{6}{7}\cdot...\cdot\dfrac{100}{101}\)

=>\(B^2< \dfrac{2}{3}\cdot\dfrac{4}{5}\cdot...\cdot\dfrac{100}{101}\cdot\dfrac{1}{2}\cdot\dfrac{3}{4}\cdot...\cdot\dfrac{99}{100}< \dfrac{1}{10}\)