Question

T = 1 x 2 x 3 + 2 x 3 x 4 + ... + 20 x 21 x 22

Answer 1

\(T=1\cdot2\cdot3+2\cdot3\cdot4+...+20\cdot21\cdot22\)

\(4T=1\cdot2\cdot3\cdot4+2\cdot3\cdot4\cdot4+...+20\cdot21\cdot22\cdot4\)

\(4T=1\cdot2\cdot3\cdot\left(4-0\right)+2\cdot3\cdot4\cdot\left(5-1\right)+..+20\cdot21\cdot22\cdot\left(23-19\right)\)

\(4T=0\cdot1\cdot2\cdot3-1\cdot2\cdot3\cdot4+2\cdot3\cdot4\cdot5-1\cdot2\cdot3\cdot4+..+20\cdot21\cdot22\cdot23-19\cdot20\cdot21\cdot22\)

\(4T=21\cdot22\cdot23\cdot24\)

\(T=\frac{21\cdot22\cdot23\cdot24}{4}=21\cdot22\cdot23\cdot6=63756\\ k.cho.mk.nha\)

Answer 2

1.2.3+2.3.4+3.4.5+...+20.21.22

=1/4.(1.2.3.4+2.3.4.4+3.4.5.4+...+20.21.22.4)

=1/4.[1.2.3.(4-0)+2.3.4.(5-1)+3.4.5.(6-2)+...+20.21.22.(23-19)]

=1/4.(1.2.3.4+2.3.4.5-1.2.3.4+3.4.5.6-2.3.4.5+....-19.20.21.22+20.21.22.23)

=1/4.20.21.22.23

=53130