Question

Tính : Q = 1.2² + 2.3² + 3.4² + …+ 19. 20²

Answer 1

Q=1.2.(3-1)+2.3.(4-1)+3.4.(5-1)+...+19.20.(21-1)=

=(1.2.3+2.3.4+3.4.5+...+19.20.21)-(1.2+2.3+3.4+...+19.20)

Đặt

A=1.2.3+2.3.4+3.4.5+...+19.20.21

4A=1.2.3.4+2.3.4.4+3.4.5.4+...+19.20.21.4=

=1.2.3.4+2.3.4(5-1)+3.4.5.(6-2)+...+19.20.21.(22-18)=

=1.2.3.4-1.2.3.4+2.3.4.5-2.3.4.5+3.4.5.6-...-18.19.20.21+19.20.21.22=

=19.20.21.22

\(A=\dfrac{19.20.21.22}{4}=5.19.21.22\)

Đặt 

B=1.2+2.3+3.4+...+19.20

3B=1.2.3+2.3.3+3.4.3+...+19.20.3=

=1.2.3+2.3.(4-1)+3.4.(5-2)+...+19.20.(21-18)=

=1.2.3-1.2.3+2.3.4-2.3.4+3.4.5-...-18.19.20+19.20.21=

=19.20.21

\(B=\dfrac{19.20.21}{3}=7.19.20\)

Q=A-B