Question

A= 1/5-1/5²+1/5³-1/5⁴+...+1/5⁹⁹-1/5¹⁰⁰

 

B= 1.2+2.3+3.4+...+2017.2018

Answer 1

a) \(A=\frac{1}{5}-\frac{1}{5^2}+\frac{1}{5^3}-\frac{1}{5^4}+...+\frac{1}{5^{99}}-\frac{1}{5^{100}}\)

\(\Rightarrow5A=1-\frac{1}{5}+\frac{1}{5^2}-\frac{1}{5^3}+...+\frac{1}{5^{98}}-\frac{1}{5^{99}}\)

\(\Rightarrow5A+A=1-\frac{1}{5^{100}}\)

\(A=\frac{1-\frac{1}{5^{100}}}{6}\)

b) B = 1.2+2.3+3.4+...+2017.2018

=>3B=1.2.3 + 2.3.3+3.4.3+...+2017.2018.3

3B = 1.2.3 + 2.3.(4-1) +3.4.(5-2) +...+2017.2018.(2019-2016)

3B = 1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+2017.2018.2019-2016.2017.2018

3B = 2017.2018.2019

\(B=\frac{2017.2018.2019}{3}\)

Answer 2

3B = 1.2.3 + 2.3.3 + 3.4.3 + ... + 2017.2018.3

3B = 1.2.3 + 2.3.(4-1) + 3.4.(5-2)+...+ 2017.2018(2019-2016)

3B = 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + ... + 2017.2018.2019 - 2016.2017.2018

3B = 2017.2018.2019

B = 2017.2018.2019/3 

B= 2739315938