S=1.2+2.3+...+n(n+1)
=>3S=1.2.3+2.3.4+...+n(n+1)3
=1.2.3+2.3(4-1)+...+n(n+1)(n+2-(n-1))
=1.2.3+2.3.4-1.2.3+...+(n-1)n(n+1)-(n-1)n(n+1)+n(n+1)(n+2)
=n(n+1)(n+2)
=>S=\(=\frac{n\left(n+1\right)\left(n+2\right)}{3}\)
vậy \(S=\frac{n\left(n+1\right)\left(n+2\right)}{3}\)
Tính tổng :
a) \(A=\frac{5}{2\cdot1}+\frac{4}{1\cdot11}+\frac{3}{11\cdot14}+\frac{1}{14\cdot15}+\frac{13}{15\cdot28}\)
b) \(B=\frac{-1}{20}+\frac{-1}{30}+\frac{-1}{42}+\frac{-1}{56}+\frac{-1}{72}+\frac{-1}{90}\)
c) \(C=\frac{1}{1\cdot2\cdot3}+\frac{1}{2\cdot3\cdot4}+...+\frac{1}{n\left(n+1\right)\left(n+2\right)}\)
d) \(D=\frac{1}{1\cdot2\cdot3\cdot4}+\frac{1}{2\cdot3\cdot4\cdot5}+...+\frac{1}{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}\)
e) \(E=\left(\frac{1}{1\cdot2\cdot3}+\frac{1}{2\cdot3\cdot4}+...+\frac{1}{37\cdot38\cdot39}\right)\cdot1482\cdot185\cdot8\)
\(\frac{1}{1\cdot2\cdot3}+\frac{1}{2\cdot3\cdot4}+\frac{1}{3\cdot4\cdot5}+.....+\frac{1}{98\cdot99\cdot100}=\frac{1}{k}\cdot\left(\frac{1}{1\cdot2}-\frac{1}{99\cdot100}\right)\)
Số k trong đẳng thức trên có giá trị là ?
Tìm y :
\(\left(\frac{1}{1\cdot2\cdot3}+\frac{1}{2\cdot3\cdot4}+\frac{1}{3\cdot4\cdot5}+...+\frac{1}{98\cdot99\cdot100}\right)\cdot y=\frac{49}{200}\)
N=\(\frac{5\cdot\left(2^2\cdot3^2\right)\cdot\left(2^2\right)^6-2\cdot\left(2^2\cdot3\right)^{14}\cdot3^6}{5\cdot2^{28}\cdot3^{19}-7\cdot2^{29}\cdot2^{18}}\)
Tinh N
Chứng minh: \(1\cdot2\cdot3+2\cdot3\cdot4+...+n\left(n+1\right)\left(n+2\right)=\frac{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}{4}\) với mọi \(n\inℕ\)
tính giá trị biểu thức ( tính nhanh nếu có thể )
c, \(1\cdot2\cdot3....9-1\cdot2\cdot3....8-1\cdot2\cdot3....7\cdot8^2\)
d,\(\frac{\left(3\cdot4\cdot2^{16}\right)^2}{11\cdot2^{13}\cdot4^{11}-16^9}\)
Tính nhanh :
a/ \(\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{9\cdot10}\).
b/ \(\left(1\cdot2\right)^{-1}+\left(2\cdot3\right)^{-1}+\left(3\cdot4\right)^{-1}+...+\left(9\cdot10\right)^{-1}\).
Tính tổng:
\(\left(1+\dfrac{1}{1\cdot3}\right)\cdot\left(1+\dfrac{1}{2\cdot4}\right)\cdot\left(1+\dfrac{1}{3\cdot5}\right)\cdot...\cdot\left(1+\dfrac{1}{2019\cdot2021}\right)\)
Tìm số tự nhiên x biết:
\(\left(\frac{1}{1\cdot2\cdot3}+\frac{1}{2\cdot3\cdot4}+...+\frac{1}{8\cdot9\cdot10}\right)\cdot x=\frac{23}{45}\)
