Ta thấy:
1.4 = 1.(1 + 3) = 1.(1 + 1 + 2) = 1.(1 + 1)+ 2.1
2.5 = 2.(2 + 3) = 2.(2 + 1 + 2) = 2.(2 + 1)+ 2.2
3.6 = 3.(3 + 3) = 3.(3 + 1 + 2) = 3.(3 + 1)+ 2.3
4.7 = 4.(4 + 3) = 4.(4 + 1 + 2) = 4.(4 + 1)+ 2.4
. . . . . . . . . . .
n(n + 3) = n(n + 1) + 2n
Vậy C = 1.2 + 2.1 + 2.3 + 2.2 + 3.4 + 2.3 + . . . + n(n + 1) + 2n
= 1.2 + 2 +2.3 + 4 + 3.4 + 6 + . . . + n(n + 1) + 2n
= [1.2 +2.3 +3.4 + . . . + n(n + 1)] + (2 + 4 + 6 + . . . + 2n)
Mà 1.2 + 2.3 + 3.4 + … + n.(n + 1) =\(\frac{n.\left(n+1\right).\left(n+2\right)}{3}\)
Và 2 + 4 + 6 + . . . + 2n =\(\frac{\left(2n+2\right).n}{2}\)
=> C=\(\frac{n.\left(n+1\right).\left(n+2\right)}{3}+\frac{\left(2n+2\right).n}{2}-\frac{n.\left(n+1\right).\left(n+5\right)}{3}\)
hok tốt
Ta có :
\(C=1.4+2.5+3.6+...+n\left(n+3\right)\)
\(\Rightarrow C=1\left(2+2\right)+2\left(3+2\right)+3\left(4+2\right)+...+n\left(n+1+2\right)\)
\(\Rightarrow C=1.2+1.2+2.3+2.2+3.4+3.2+...+n\left(n+1\right)+n.2\)
\(\Rightarrow C=\left(1.2+2.3+3.4+...+n\left(n+1\right)\right)+2\left(1+2+3+...+n\right)\)
\(\Rightarrow C=\frac{n\left(n+1\right)\left(n+2\right)}{3}+2\left(\frac{\left(n+1\right).n}{2}\right)\)
\(\Rightarrow C=\frac{n\left(n+1\right)\left(n+2\right)}{3}+\left(n+1\right)n\)
~
C = 1.4+2.5+3.6+4.7+...+n.(n+3)
C= 1.(2+2) + 2.(3+2) + 3.(4+2) + 4.(5+2) + ...+n.[(n+1) + 2]
C = 1.2 + 1.2 + 2.3 + 2.2 + 3.4 + 3.2 + 4.5 + 4.2 + ...+ n.(n+1) + n.2
C = [(1.2+2.3+3.4+4.5+...+n.(n+1)] + ( 1.2+2.2+3.2+4.2+...+n.2)
Đặt A = 1.2 + 2.3 + 3.4+4.5 + ...+n.(n+1)
=>3A = 1.2.3+2.3.3+3.4.3+4.5.3+...+n.(n+1).3
3A = 1.2.(3-0)+2.3.(4-1) + 3.4.(5-2) + ...+n.(n+1).[(n+2) - (n-1)]
3A = 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 -2.3.4 + n.(n+1).(n+2) - (n-1).n.(n+1)
3A = n.(n+1).(n+2)
\(A=\frac{n.\left(n+1\right).\left(n+2\right)}{3}\)
Đặt B = 1.2+2.2+3.2+4.2 +...+n.2
B = 2.(1+2+3+4+...+n)
B = 2. [(1+n).n:2]
B = 2. (1+n).n . 1/2
B = (1+n).n
Thay A;B vào C
có: \(C=\frac{n.\left(n+1\right).\left(n+2\right)}{3}+\left(1+n\right).n\)