(+) \(S=1+2+3+...+200\)
\(\Rightarrow S=\frac{\left(200+1\right)200}{2}=20100\)
(+) \(A=1.2+2.3+....+199.200\)
\(\Rightarrow3A=1.2.\left(3-0\right)+2.3.\left(4-1\right)+.....+199.200.\left(201-198\right)\)
\(\Rightarrow3A=1.2.3-0.1.2+2.3.4-1.2.3+.....+199.200.201-198.199.200\)
\(\Rightarrow3A=199.200.201\)
\(\Rightarrow A=\frac{199.200.201}{3}\)
\(\Rightarrow A=2666600\)
(+) \(B=1^2+2^2+....+200^2\)
\(\Rightarrow B=1\left(0+1\right)+2\left(1+1\right)+3\left(2+1\right)+....+200\left(199+1\right)\)
\(\Rightarrow B=1+1.2+2+2.3+3+.....+199.200+200\)
\(\Rightarrow B=\left(1+2+3+....+200\right)+\left(1.2+2.3+....+199.200\right)\)
\(\Rightarrow B=S+A\)
\(\Rightarrow B=2666600+20100=2686700\)
S = 1 + 2 + 3 + ... + 200
S = (1 + 200).200:2
S = 201.100
S = 20100
A = 1.2 + 2.3 + ... + 199.200
3A = 1.2.(3-0) + 2.3.(4-1) + ... + 199.200.(201-198)
3A = 1.2.3 - 0.1.2 + 2.3.4 - 1.2.3 + ... + 199.200.201 - 198.199.200
3A = 199.200.201
A = 199.200.67
A = 2666600
B = 12 + 22 + 32 + ... + 2002
B = 1.(0 + 1) + 2.(1 + 1) + 3.(2 + 1) + ... + 200.(199 + 1)
B = 0.1 + 1.1 + 1.2 + 1.2 + 2.3 + 1.3 + ... + 199.200 + 1.200
B = (0.1 + 1.2 + 2.3 + ... + 199.200) + (1.1 + 1.2 + 1.3 + ... + 1.200)
B = 2666600 + (200 + 1).200:2
B = 2666600 + 201.100
B = 2666600 + 20100
B = 2686700
