Question

S= 1*2+2*3+3*4+...+38*39+39*40

S= ?

Answer 1

\(3S=1\cdot2\cdot3+2\cdot3\cdot\left(4-1\right)+...+39\cdot40\cdot\left(41-38\right)\)

\(3S=1\cdot2\cdot3+2\cdot3\cdot4-1\cdot2\cdot3+...+39\cdot40\cdot41-38\cdot39\cdot40\)

\(3S=39\cdot40\cdot41\)

\(=>S=39\cdot40\cdot41:3=21320\)

Answer 2

  S=1.2 + 2.3 + 3.4 +...+ 38.39 + 39.40

3S=1.2.3 + 2.3.3 +3.4.3 +...+ 38.39.3 +39.40.3

3S=1.2.(3-0) + 2.3.(4-1) + 3.4.(5-2) +...+ 38.39.(40-37) + 39.40.(41-38)

3S=1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 +...+ 38.39.40 - 37.38.39 + 39.40.41 - 38.39.40

3S=(1.2.3 - 1.2.3)+(2.3.4 - 2.3.4)+(3.4.5 - 3.4.5)+...+(37.38.39 - 37.38.39)+(38.39.40 - 38.39.40)+39.40.41

3S=39.40.41

  S=\(\frac{39.40.41}{3}\) 

  \(\Rightarrow\)S=21320