Question 4
Evaluate:
a) $A=\frac1{1\times2}+\frac1{2\times3}+\frac1{3\times4}+\cdots+\frac1{2020\times2021}$.
b) $B=\frac16+\frac1{12}+\frac1{20}+\frac1{30}+\cdots+\frac1{2070}$.
c) $C=\frac1{1\times3}+\frac1{3\times5}+\frac1{5\times7}+\cdots+\frac1{2019\times2021}$.
d) $D=1+\frac1{1+2}+\frac1{1+2+3}+\frac1{1+2+3+4}+\cdots+\frac1{1+2+\cdots+2020}$.
e) $E=1+\frac12+\frac14+\frac18+\frac1{16}+\cdots+\frac1{2048}$.
f) $F=1+\frac13+\frac19+\frac1{27}+\frac1{81}+\cdots+\frac1{2187}$.
**Editorial note for this self-study exercise:** Continue the displayed rules: consecutive products in (a),(b), consecutive odd products in (c), triangular-number denominators in (d), and geometric ratios 1/2 and 1/3 in (e),(f). Enter all six answers in order a,b,c,d,e,f, each as a reduced positive fraction p/q, separated by semicolons with no spaces. Format-only example: 1/2;2/3;3/4;4/5;5/6;6/7.
a) $A=\frac1{1\times2}+\frac1{2\times3}+\frac1{3\times4}+\cdots+\frac1{2020\times2021}$.
b) $B=\frac16+\frac1{12}+\frac1{20}+\frac1{30}+\cdots+\frac1{2070}$.
c) $C=\frac1{1\times3}+\frac1{3\times5}+\frac1{5\times7}+\cdots+\frac1{2019\times2021}$.
d) $D=1+\frac1{1+2}+\frac1{1+2+3}+\frac1{1+2+3+4}+\cdots+\frac1{1+2+\cdots+2020}$.
e) $E=1+\frac12+\frac14+\frac18+\frac1{16}+\cdots+\frac1{2048}$.
f) $F=1+\frac13+\frac19+\frac1{27}+\frac1{81}+\cdots+\frac1{2187}$.
**Editorial note for this self-study exercise:** Continue the displayed rules: consecutive products in (a),(b), consecutive odd products in (c), triangular-number denominators in (d), and geometric ratios 1/2 and 1/3 in (e),(f). Enter all six answers in order a,b,c,d,e,f, each as a reduced positive fraction p/q, separated by semicolons with no spaces. Format-only example: 1/2;2/3;3/4;4/5;5/6;6/7.