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Đặt A = 1.2.3 + 2.3.4 + 3.4.5 + ... + 28.29.30
4A = 1.2.3.(4-0) + 2.3.4.(5-1) + 3.4.5.(6-2) + ... + 28.29.30.(31-27)
4A = 1.2.3.4 - 0.1.2.3. + 2.3.4.5 - 1.2.3.4 + 3.4.5.6 - 2.3.4.5 + ... + 28.29.30.31 - 27.28.29.30
4A = 28.29.30.31 - 0.1.2.3
4A = 28.29.30.31
\(A=\frac{28.29.30.31}{4}=7.29.30.31=188790\)
Theo cách tính trên ta dễ dàng tính được:
1.2.3 + 2.3.4 + 3.4.5 + ... + (n - 1).n.(n + 1) = \(\frac{\left(n-1\right).n.\left(n+1\right).\left(n+2\right)}{4}\)
Từ công thức \(\frac{2}{a\left(a+1\right)\left(a+2\right)}=\frac{1}{a\left(a+1\right)}-\frac{1}{\left(a+1\right)\left(a+2\right)}\), ta có:
\(2C=\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{a\left(a+1\right)\left(a+2\right)}\)
\(2C=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{a\left(a+1\right)}-\frac{1}{\left(a+1\right)\left(a+2\right)}\)
\(2C=\frac{1}{1.2}-\frac{1}{\left(a+1\right)\left(a+2\right)}\)
\(C=\left[\frac{1}{2}-\frac{1}{\left(a+1\right)\left(a+2\right)}\right]:2=\frac{\left(a+1\right)\left(a+2\right)-2}{4\left(a+1\right)\left(a+2\right)}=\frac{a\left(a+3\right)}{4\left(a+1\right)\left(a+2\right)}\)
=1+\(\frac{1}{2}\)-\(\frac{1}{3}\)+\(\frac{1}{2}\) -\(\frac{1}{3}\) -\(\frac{1}{4}\)+\(\frac{1}{3}\) - \(\frac{1}{4}\)-\(\frac{1}{5}\)+.....+\(\frac{1}{99}\)-\(\frac{1}{100}\)-\(\frac{1}{101}\)
=1+\(\frac{1}{101}\)
=\(\frac{102}{101}\)
1/1.2.3 = 1/2 .[1/1.2 - 1 / 2.3]
1/2.3.4 = 1/2[ 1/2- 1/3 ]
...................
1/99.100.101 = 1/2[ 1/99. 100 - 1/100.101]
=> A= 1/2 [ 1/1.2- 1/2.3 + 1/2.3 - 1/3.4 + 1/3.4 - 1/ 4.5 +.........+ 1/99 .100 - 1/100. 101]
A = 1/2 . [1/1.2 -1/100 .101]
A= 1/2 . 5049 /10100 = 5049 / 20200.
Mình nghĩ là vậy đó.
1/1.2.3+1/2.3.4+1/3.4.5+...+1/37.38.39
= 1/2.(1/1.2-1/2.3)+1/2.(1/2.3-1/3.4)+...+1/2.(1/37.38-1/38.39)
= 1/2.(1/1.2-1/2.3+1/2.3-1/3.4+...+1/37.38-1/38.39)
= 1/2.(1/1.2-1/38.39)
= 1/2.370/741
= 185/741
Đặt S = 1/1.2.3 - 1/2.3.4 - 1/3.4.5 - ...- 1/97.98.99
S x 2 = 2/1.2.3 - 2/2.3.4 - 2/3.4.5 - ...- 2/97.98.99
= (1/1.2 -1/2.3) - (1/2.3 - 1/3.4 ) - (1/3.4 - 1/4.5) - ...- (1/97.98 - 1/98.99)
= 1/1.2 - 1/2.3 - 1/2.3 + 1/3.4 - 1/3.4 + 1/4.5 - ....- 1/97.98 + 1/98.99
= 1/2 -1/3 + 1/98.99
= 1618/9072 => S = 1618/9072 : 2 = 809/9072
Đặt A=1/1.2.3+1/2.3.4+...+1/99.100.101
2A=2/1.2.3+2/2.3.4+...2/99.100.101
2A=3-1/1.2.3+4-2/2.3.4+...+101-99/99.100.101
2A=3/1.2.3-1/1.2.3+4/2.3.4-2/2.3.4+...+101/99.100.101-99/99.100.101
2A=1/1.2-1/2.3+1/2.3-1/3.4+...+1/99.100-1/100.101
2A=1/2-1/10100
