Question

Tính nhanh  :

a ) S = 2+4+6+8+.....+2018

b ) S= 10+10²+10³+...+10¹⁰⁰

c) S=\(\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+.....+\frac{1}{5^{100}}\)

d) S=\(\frac{1!}{3!}+\frac{2!}{4!}+\frac{3!}{5!}+....+\frac{2018!}{2020!}\)

biết rằng : n!=1×2×3×...×n

VD : 1! = 1

2! = 2.1

3!=1.2.3

4!=1.2.3.4

! : giai thừa

Answer 1

a) 2 +4+6+8+...+2018

= ( 2018+2) x 1009 : 2

= 2020 x 1009 : 2

= 1009 x (2020:2)

= 1009 x 1010

= 1 019 090

b) S = 10 + 10² + 10³ + ...+ 10¹⁰⁰

=> 10.S = 10² + 10³ + 10⁴ +...+ 10¹⁰¹

=> 10.S - S = 10¹⁰¹-10

9.S=10¹⁰¹- 10

\(\Rightarrow S=\frac{10^{101}-10}{9}\)

c) \(S=\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}+...+\frac{1}{5^{100}}\)

\(\Rightarrow5S=1+\frac{1}{5}+\frac{1}{5^2}+...+\frac{1}{5^{99}}\)

\(5S-S=1-\frac{1}{5^{100}}\)

\(4S=1-\frac{1}{5^{100}}\)

\(S=\frac{1-\frac{1}{5^{100}}}{4}\)

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Answer 2

d) \(S=\frac{1!}{3!}+\frac{2!}{4!}+\frac{3!}{5!}+...+\frac{2018!}{2020!}\)

\(S=\frac{1}{1.2.3}+\frac{1.2}{1.2.3.4}+\frac{1.2.3}{1.2.3.4.5}+...+\frac{1.2.3...2018}{1.2.3...2020}\)

\(S=\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{2019.2020}\)

\(S=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{2019}-\frac{1}{2020}\)

\(S=\frac{1}{2}-\frac{1}{2020}\)

\(S=\frac{1009}{2020}\)