Ta nhận thấy

\(\dfrac{1}{n\cdot\left(n+2\right)}-\dfrac{1}{\left(n+2\right)\cdot\left(n+4\right)}\\ =\dfrac{n+4}{n\cdot\left(n+2\right)\cdot\left(n+4\right)}-\dfrac{n}{n\cdot\left(n+2\right)\cdot\left(n+4\right)}\\ =\dfrac{n+4-n}{n\cdot\left(n+2\right)\cdot\left(n+4\right)}\\ =\dfrac{4}{n\cdot\left(n+2\right)\cdot\left(n+4\right)}\)

\(A=\dfrac{4}{2\cdot4\cdot6}+\dfrac{4}{4\cdot6\cdot8}+\dfrac{4}{6\cdot8\cdot10}+...+\dfrac{4}{46\cdot48\cdot50}\\ =\dfrac{1}{2\cdot4}-\dfrac{1}{4\cdot6}+\dfrac{1}{4\cdot6}-\dfrac{1}{6\cdot8}+\dfrac{1}{6\cdot8}-\dfrac{1}{8\cdot10}+...+\dfrac{1}{46\cdot48}-\dfrac{1}{48\cdot50}\\ =\dfrac{1}{2\cdot4}-\dfrac{1}{48\cdot50}\\ =\dfrac{1}{8}-\dfrac{1}{2400}\\ =\dfrac{300}{2400}-\dfrac{1}{2400}\\ =\dfrac{299}{2400}\)

Số nghịch đảo của \(A\) là \(\dfrac{2400}{299}\)

Ta có :

B = 2 . 4 . 6 + 4 . 6 .8 + ......+ 96 .98 .100

B.8 = 2 . 4 . 6 . 8 + 4. 6 . 8 .( 1 0 - 2) +........+96 . 98 . 100 . ( 102 - 94 )

8B = 2 . 4 . 6 .8 + 4. 6 .8 . 10 - 2 . 4 . 6 . 8 +......+ 96 . 98 . 100. 102 - 94 . 96 . 98 . 100

8B = 96 . 98 . 100 . 102

B = 96 . 98 . 100 .1 02 : 8

B = 12 . 98 . 100. 102

S = 2.4.6 + 4.6.8 + ... + 98.100.102

=> 8S = 2.4.6.8 + 4.6.8.8 + ... + 98.100.102.8

=> 8S = 2.4.6.(8 - 0) + 4.6.8.(10 - 2) + ... + 98.100.102.(104 - 96)

=> 8S = 2.4.6.8 - 0 + 4.6.8.10 - 2.4.6.8 + ... + 98.100.102.104 - 96.98.100.102

=> 8S = 98.100.102.104

=> S = 98.100.102.104/8

=> S = 12994800

=> 8S = 2.4.6.8 + 4.6.8.8 + 6.8.10.8 + .... + 98.100.102.8

=> 8S = 2.4.6.8 + 4.6.8.( 10 - 2 ) + 6.8.10.( 12 - 4 ) + .... + 98.100.102.( 104 - 96 )

=> 8S = 2.4.6.8 + 4.6.8.10 - 2.4.6.8 + 6.8.10.12 - 4.6.8.10 + .... + 98.100.102.104 - 96.98.100.102

=> 8S = ( 2.4.6.8 - 2.4.6.8 ) + ( 4.6.8.10 - 4.6.8.10 ) + .... + ( 96.98.100.102 - 96.98.100.102 ) + 98.100.102.104

=> 8S = 98.100.102.104

=> S = \(\frac{98.100.102.104}{8}\)