Question

Cho biểu thức A=1+x+x²+...+x⁹⁹+x¹⁰⁰

hãy chứng minh A=\(\frac{x^{101}-1}{x-1}\)

 

Answer 1

Suy ra xA=x+x^2+x^3+...+x^101

xA-A=x^101-1

A(x-1)=x^101-1

A=(x^101-1)/(x-1)

Answer 2

\(A=1+x+x^2+...+x^{99}+x^{100}\Rightarrow x.A=x+x^2+x^3+...+x^{100}+x^{101}\)

\(\Rightarrow x.A-A=\left(x+x^2+x^3+...+x^{100}+x^{101}\right)-\left(1+x+x^2+...+x^{99}+x^{100}\right)\)

\(\Rightarrow\left(x-1\right).A=x^{101}-1\Rightarrow A=\frac{x^{101}-1}{x-1}\) (đpcm)