chứng minh

\(\frac{1}{n}\cdot\frac{1}{n+4}=\frac{1}{4}\cdot\left(\frac{1}{n}-\frac{1}{n+4}\right)\)

Tính các tích sau: với n là số tự nhiên, n<3

a) \(\left(1-\frac{1}{2}\right)\cdot\left(1-\frac{1}{3}\right)\cdot\left(1-\frac{1}{4}\right)\cdot...\cdot\left(1-\frac{1}{n}\right)\)

b) \(\left(1-\frac{1}{2^2}\right)\cdot\left(1-\frac{1}{3^2}\right)\cdot\left(1-\frac{1}{4^2}\right)\cdot...\cdot\left(1-\frac{1}{n^2}\right)\)

Chứng minh:
a, \(\left(1+\frac{1}{1\cdot3}\right)\left(1+\frac{1}{2\cdot4}\right)\left(1+\frac{1}{3\cdot5}\right)\cdot...\cdot\left(1+\frac{1}{n\left(n+2\right)}\right)< 2\)
b, \(\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}< \frac{5}{4}\)

Chứng minh

\(\frac{2}{\left(n-1\right)\cdot n\left(n+1\right)}=\frac{1}{n\cdot\left(n-1\right)}-\frac{1}{n\cdot\left(n+1\right)}\)

Chứng minh rằng

\(\frac{1\cdot3\cdot5\cdot\cdot\cdot\left(2n-1\right)}{\left(n+1\right)\cdot\left(n+2\right)\cdot\left(n+3\right)\cdot...\cdot2n}=\frac{1}{2^n}\)

Chứng minh rằng:

a)\(\frac{1\cdot3\cdot5\cdot\cdot\cdot39}{21\cdot22\cdot23\cdot\cdot\cdot40}=\frac{1}{2^{20}}\)

b)\(\frac{1\cdot3\cdot5\cdot\cdot\cdot\left(2n-1\right)}{\left(n+1\right)\left(n+2\right)\left(n+3\right)\cdot\cdot\cdot2n}=\frac{1}{2^n}\)Với \(n\inℕ^∗\)

Chứng minh rằng : 

\(\frac{1\cdot3\cdot5\cdot...\cdot\left(2n-1\right)}{\left(n+1\right)\cdot\left(n+2\right)\cdot\left(n+3\right)\cdot...\cdot2n}=\frac{1}{2^n}\)

chứng minh   \(y=\frac{1}{\left(n+1\right)\cdot\sqrt{n}+n\cdot\sqrt{n+1}}=\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\)

Tính:

a) M=\(\frac{2\cdot2012}{1+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+...+\frac{1}{1+2+3+...+20012}}\)

b) N= \(1+\frac{1}{2\cdot\left(1+2\right)}+\frac{1}{3\cdot\left(1+2+3\right)+}+\frac{1}{4}\cdot\left(1+2+3+4\right)\)\(+...+\frac{1}{16}\cdot\left(1+2+3+4+...+16\right)\)

Giúp nha mình tick