Question

a)Rút gọn biểu thức A=\(1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2017}}\)

b) So sánh : A=\(\frac{20^{10}+1}{20^{10}-1}\) và B=\(\frac{20^{10}-1}{20^{10}-3}\)

Answer 1

a) Ta có: \(A=\frac{2^{2017}}{2^{2017}}+\frac{2^{2016}}{2^{2017}}+\frac{2^{2015}}{2^{2017}}+...+\frac{2^1}{2^{2017}}+\frac{1}{2^{2017}}\)

\(=\frac{1+2^1+2^2+...+2^{2016}+2^{2017}}{2^{2017}}\)

Đặt: B=\(1+2^1+2^2+...+2^{2017}\)

\(\Leftrightarrow2B=2^1+2^2+2^3+....+2^{2017}+2^{2018}\)

\(\Leftrightarrow2B-B=2^{2018}-1\)

\(\Leftrightarrow B=2^{2018}-1\)

\(\Rightarrow A=\frac{B}{2^{2017}}=\frac{2^{2018}-1}{2^{2017}}\)

Mik chỉ biết làm phần a thôi

Answer 2

b/ Sử dụng quy tắc: \(\frac{a+c}{b+c}< \frac{a}{b}\) với \(\left\{{}\begin{matrix}a;b;c>0\\a>b\end{matrix}\right.\)

\(B=\frac{2^{10}-1}{2^{10}-3}>\frac{2^{10}-1+2}{2^{10}-3+2}=\frac{2^{10}+1}{2^{10}-1}\)

\(\Rightarrow B>A\)