Question

Tính:

a,A=1.2+2.3+3.4+......+99.100

b,B=1.3+2.4+3.5+......+99.101

c,C=1.4+2.5+3.6+......+99.102

d,D=1.2.3+2.3.4+3.4.5+.......+98.99.100

 

Answer 1

\(A=1.2+2.3+3.4+...+99.100\)

\(\Rightarrow3A=1.2.3+2.3\left(4-1\right)+3.4\left(5-2\right)+...+90.100\left(101-98\right)\)

\(\Rightarrow3A=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+99.100.101-98.99.100\)

\(\Rightarrow3A=99.100.101\)

\(\Rightarrow A=\left(99.100.101\right):3\)

\(\Rightarrow A=333300\)

Answer 2

\(B=1.3+2.4+3.5+...+99.101\)

\(\Rightarrow B=1\left(2+1\right)+2\left(3+1\right)+3\left(4+1\right)+...+99\left(100+1\right)\)

\(\Rightarrow B=1.2+1+2.3+2+3.4+3+...+99.100+99\)

\(\Rightarrow B=\left(1.2+2.3+3.4+...+99.100\right)+\left(1+2+3+...+99\right)\)

\(\Rightarrow B=333300+4950\)

\(\Rightarrow B=338250\)

Answer 3

\(D=1.2.3+2.3.4+3.4.5+...+98.99.100\)

\(\Rightarrow4D=1.2.3.4+2.3.4.4+...+98.99.100.4\)

\(\Rightarrow4D=1.2.3\left(4-0\right)+2.3.4\left(5-1\right)+...+98.99.100\left(101-97\right)\)

\(\Rightarrow4D=1.2.3.4-0.1.2.3+2.3.4.5-1.2.3.4+...+98.99.100-97.98.99\)

\(\Rightarrow4D=98.99.100\)

\(\Rightarrow D=\left(98.99.100\right):4=242550\)

Answer 4

\(C=1.4+2.5+3.6+...+99.102\)

\(=1\left(2+2\right)+2\left(3+2\right)+3\left(4+2\right)+...+99\left(100+2\right)\)

\(=1.2+1.2+2.3+2.2+3.4+3.2+...+99.100+99.2\)

\(=\left(1.2+2.3+3.4+...+99.100\right)+2\left(1+2+3+...+99\right)\)

Đặt \(C=1.2+2.3+3.4+...+99.100\)

\(\Rightarrow3C=1.2\left(3-0\right)+2.3\left(4-1\right)+3.4\left(5-2\right)+...+99.100\left(101-98\right)\)

\(\Rightarrow3C=1.2.3-0.1.2+2.3.4-1.2.3+3.4.5-2.3.4+...+99.100.101-98.99.100\)

\(\Rightarrow3C=99.100.101-0.1.2\)

\(\Rightarrow3C=999900:0\)

\(\Rightarrow C=333300\)

\(\left(1.2+2.3+3.4+...+99.100\right)+2\left(1+2+3+...+99\right)\)

\(=333300+2.4950\)

\(=333300+9900\)

\(=343200\)