Question

Cho \(A=1+2012+2012^2+2012^3+......+2012^{100}\)

\(B=2012^{101}:2011\)

Tính \(B-A\)

Answer 1

Ta có :

\(A=1+2012+2012^2+2012^3+........+2012^{100}\)

\(2012A=2012+2012^2+2012^3+......+2012^{100}+2012^{101}\)

\(\Rightarrow2012A-A=\left(2012+2012^2+.......+2012^{101}\right)-\left(1+2012+........+2012^{100}\right)\)

\(2011A=2012^{101}-1\)

\(\Rightarrow A=\dfrac{2012^{101}-1}{11}\)

Mà \(B=\dfrac{2012^{101}}{2011}\)

\(\Rightarrow B-A=\dfrac{2012^{101}}{2011}-\dfrac{2012^{101}-1}{2011}\)

\(=\dfrac{2012^{101}-\left(2012^{101}-1\right)}{2011}\)

\(=\dfrac{2012^{101}-2012^{101}+1}{2011}\)

\(=\dfrac{1}{2011}\)

~ Chúc bn học tốt ~

Answer 2

Có: \(2012A=2012+2012^2+...+2012^{101}\)
=> \(2012A-A=\left(2012+2012^2+...+2012^{101}\right)-\left(1+2012+...+2012^{100}\right)\)
\(\Rightarrow2011A=2012^{101}-1\)
\(\Rightarrow A=\dfrac{2012^{101}-1}{2011}\)
Do đó \(B-A=\dfrac{2012^{101}}{2011}-\dfrac{2012^{101}-1}{2011}=\dfrac{1}{2011}\)