Question

Tính hợp lí:

a) \(\frac{1+2+2^2+2^3+...+2^{2009}}{1-2^{2010}}\)

b) \(\frac{1}{299\cdot297}\) - \(\frac{1}{297\cdot295}\) - \(\frac{1}{295\cdot293}\) - ... -\(\frac{1}{3\cdot1}\)

c) \(\frac{12}{1\cdot3\cdot5}\)+\(\frac{12}{3\cdot5\cdot7}\)+\(\frac{12}{5\cdot7\cdot9}\)+...+\(\frac{12}{25\cdot27\cdot29}\)

 

Answer 1

a) Đặt A= \(\frac{1+2+2^2+2^3+...+2^{2009}}{1-2^{2010}}\)

Đặt S = 1 + 2 + 2² + 2³ + ... + 2²⁰⁰⁹

=> 2S = 2 + 2² + 2³ + ... + 2²⁰¹⁰

=> 2S - S = (2 + 2² + 2³ + ... + 2²⁰¹⁰) - (1 + 2 + 2² + 2³ + .. + 2²⁰⁰⁹)

=> S = 2²⁰¹⁰ - 1

=> S = - 1  - 2²⁰¹⁰

^{\(\Rightarrow A=\frac{-1-2^{2010}}{1-2^{2010}}=-1\)}

Vậy \(\frac{1+2+2^2+2^3+...+2^{2009}}{1-2^{2010}}=-1\)

b) Đặt: \(A=\frac{1}{299.297}-\frac{1}{297.295}-\frac{1}{295.293}-...-\frac{1}{3.1}\)

\(\Rightarrow-2A=-\frac{2}{299.297}+\frac{2}{297.295}+\frac{2}{295.293}+...+\frac{2}{3.1}\)

\(\Rightarrow-2A=\frac{2}{1.3}+\frac{2}{3.5}+...+\frac{2}{295.297}-\frac{2}{297.299}\)

\(\Rightarrow-2A=1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{295}-\frac{1}{297}-\frac{1}{297.299}\)

\(\Rightarrow-2A=1-\frac{1}{297}-\frac{2}{88803}\)

\(\Rightarrow-2A=\frac{296}{297}-\frac{2}{88803}=\frac{88504}{88803}-\frac{2}{88803}=\frac{88502}{88803}\)

\(\Rightarrow A=\frac{88502}{88803}:\left(-2\right)=\frac{44251}{88803}\)

Vậy \(\frac{1}{299.297}-\frac{1}{297.295}-\frac{1}{295.293}-...-\frac{1}{3.1}=\frac{44251}{88803}\)

c) Đặt \(B=\frac{12}{1.3.5}+\frac{12}{3.5.7}+\frac{12}{5.7.9}+...+\frac{12}{25.27.29}\)

\(\Rightarrow\frac{B}{3}=\frac{4}{1.3.5}+\frac{4}{3.5.7}+\frac{4}{5.7.9}+...+\frac{12}{25.27.29}\)

\(\Rightarrow\frac{B}{3}=\frac{1}{1.3}-\frac{1}{3.5}+\frac{1}{3.5}-\frac{1}{5.7}+\frac{1}{5.7}-\frac{1}{7.9}+...+\frac{1}{25.27}-\frac{1}{27.29}\)

\(\Rightarrow\frac{B}{3}=\frac{1}{1.3}-\frac{1}{27.29}\)

\(\Rightarrow\frac{B}{3}=\frac{1}{3}-\frac{1}{783}=\frac{261}{783}-\frac{1}{783}=\frac{260}{783}\)

\(\Rightarrow B=\frac{260}{783}.3=\frac{260}{261}\)

Vậy \(\frac{12}{1.3.5}+\frac{12}{3.5.7}+\frac{12}{5.7.9}+...+\frac{12}{25.27.29}=\frac{260}{261}\)

Duyệt mk nha!!!