99^2+99*2+1=? 40^2-39^2+38^2-37^2+........+2^2-1^2 2019^2-2018*2020

a)992+2.99+1=(99+1)2=1002=10000.b)402−392+382−372+.....+22−12=(402−392)+(382−372)+....+(22−12)=(40−39)(40+39)+(38−37)(38+37)+....+(2−1)(2+1)=40+39+38+37+....+2+1=40(40+1)2=20.41=820.c)20192−20182=(2019−2018)(2019+2018)=4037

\[\begin{array}{l} a)\,\,\,{99^2} + 2.99 + 1 = {\left( {99 + 1} \right)^2} = {100^2} = 10000.\\ b)\,\,{40^2} - {39^2} + {38^2} - {37^2} + ..... + {2^2} - {1^2}\\ = \left( {{{40}^2} - {{39}^2}} \right) + \left( {{{38}^2} - {{37}^2}} \right) + .... + \left( {{2^2} - {1^2}} \right)\\ = \left( {40 - 39} \right)\left( {40 + 39} \right) + \left( {38 - 37} \right)\left( {38 + 37} \right) + .... + \left( {2 - 1} \right)\left( {2 + 1} \right)\\ = 40 + 39 + 38 + 37 + .... + 2 + 1\\ = \frac{{40\left( {40 + 1} \right)}}{2} = 20.41 = 820.\\ c)\,\,\,{2019^2} - {2018^2} = \left( {2019 - 2018} \right)\left( {2019 + 2018} \right)\\ = 4037. \end{array}\]