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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
\(\dfrac{2.3.5.\left(-7\right).12}{3.\left(-5\right).7.14}=\dfrac{2.3.5.\left(-7\right).12}{3.5.\left(-7\right).2.7}=\dfrac{12}{7}\)
\(T=\left(\frac{1}{2}+1\right)\left(\frac{1}{3}+1\right)\left(\frac{1}{4}+1\right)+...+\left(\frac{1}{99}+1\right)\)
\(T=\frac{3}{2}.\frac{4}{3}.\frac{5}{4}.....\frac{100}{99}\)
\(T=\frac{1}{2}.100\)
\(T=50\)
