Question

Cho S + 1 + 2 + 2² + 2³ + ........... + 2²⁰¹⁷

a. Thu gọn S

So sánh S và 5 . 2²⁰¹⁶

Answer 1

a, \(S=1+2+2^2+2^3+...+2^{2017}\)

\(\Rightarrow2S=2+2^2+2^3+2^4+...+2^{2018}\)

\(\Rightarrow2S-S=\left(2+2^2+2^3+...+2^{2018}\right)-\left(1+2+2^2+...+2^{2017}\right)\)

\(\Rightarrow S=2^{2018}-1\)

Vậy : \(S=2^{2018}-1\)

b, Ta có : \(2^{2018}-1< 2^{2018}=2^2.2^{2016}=4.2^{2016}< 5.2^{2016}\)

Vì : \(2^{2018}-1< 4.2^{2016}< 5.2^{2016}\Rightarrow S< 5.2^{2016}\)

Vậy : \(S< 5.2^{2016}\)