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Bài làm:
Ta có: \(\frac{x}{1.2}+\frac{x}{2.3}+\frac{x}{3.4}+...+\frac{x}{2017.2018}=-1\)
\(\Leftrightarrow x\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2017.2018}\right)=-1\)
\(\Leftrightarrow x\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2017}-\frac{1}{2018}\right)=-1\)
\(\Leftrightarrow x\left(1-\frac{1}{2018}\right)=-1\)
\(\Leftrightarrow x.\frac{2017}{2018}=-1\)
\(\Rightarrow x=-\frac{2018}{2017}\)
P=1x2+2x3+3x4+...+2017x2018
3P = 1x2x3 + 2x3x3 + 3x4x3 + ... + 2017x2018x3
3P = 1x2x3 + 2x3x(4-1) + 3x4x(5-2) + ... +2017x2018x(2019-2016)
3P = 1x2x3 + 2x3x4 - 1x2x3 + 3x4x5 - 2x3x4 + ... + 2017x2018x2019 - 2016x2017x2018
3P = 2017x2018x2019
P = 2017x2018x2019 : 3
P = 2739315938
P = 1x2+2x3+3x4+...+2017x2018
3xP = 1x2x3+2x3x3+3x4x3+...+2017x2018x3
3xP = 1x2x3+2x3x(4-1)+3x4x(5-2)+...+2017x2018x(2019-2016)
3xP = 1x2x3+2x3x4-2x3x1+3x4x5-3x4x2+...+2017x2018x2019-2017x2018x2016
3xP = 2017x2018x2019
3xP = 8217947814
P = 8217947814 : 3
P = 2739315938
\(S=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2017}-\frac{1}{2018}\)
\(S=1-\frac{1}{2018}\)
\(S=\frac{2018}{2018}-\frac{1}{2018}\)
\(S=\frac{2017}{2018}\)
\(S=\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{2017.2018}.\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-...-\frac{1}{2017}+\frac{1}{2017}-\frac{1}{2018}\)
\(=1-\frac{1}{2018}=\frac{2017}{2018}\)
S = 1.2+2.3+3.4+....+38.39+39.40
S.3 = ( 1.2+2.3+3.4+...+38.39+39.40 ) . 3
S.3 = 1.2.(3-0)+2.3.(4-1)+3.4.(5-2)+...........+38.39.(40-37)+39.40.(41-38)
S.3 = 1.2.3-0.1.2+2.3.4-1.2.3+3.4.5-2.3.4+...........+38.39.40-37.38.39+39.40.41-38.39.40
S = \(\frac{39.40.41}{3}\)
S = 13.40.41
S = 21320
A = 1.2 + 2.3 + 3.4 + ... + 2017.2018
⇒ 3A = 1.2.3 + 2.3.(4 - 1) + 3.4.(5 - 2) + ... + 2017.218.(2019 - 2016)
= 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + ... + 2017.2018.2019 - 2016.2017.2018
= 2017.2018.2019
= 2017.2018.2019
B = 2018³/3 ⇒ 3B = 2018³
Ta có:
2017.2019 = (2018 - 1).(2018 + 1)
= 2018² - 1²
= 2018.2018 - 1 < 2018.2018
⇒ 2017.2018.2019 < 2018.2018.2018
⇒ 3A < 3B
⇒ A < B