\(\left(5+5^2+5^3+...+5^{10}\right)+4x-1=\frac{1}{4}5^{11}+\frac{1}{2}x+3\)
\(\Leftrightarrow\left(1+5+5^2+5^3+...+5^{10}\right)+4x-2=\frac{1}{4}5^{11}+\frac{1}{2}x+3\)(1)
1./ Trước tiên, ta tính:
\(S=1+5+5^2+5^3+...+5^{10}\)
\(\left(5-1\right)\cdot S=\left(5-1\right)\left(1+5+5^2+5^3+...+5^{10}\right)\)
\(\Leftrightarrow4S=5^{11}-5^{10}+5^{10}-5^9+...+5-1=5^{11}-1\)
\(\Leftrightarrow S=\frac{5^{11}-1}{4}=\frac{1}{4}5^{11}-\frac{1}{4}\)
2./ (1) trở thành:
\(\Leftrightarrow\frac{1}{4}5^{11}-\frac{1}{4}+4x-2=\frac{1}{4}5^{11}+\frac{1}{2}x+3\)
\(\Leftrightarrow4x-\frac{1}{2}x=5+\frac{1}{4}\Leftrightarrow\frac{7}{2}x=\frac{21}{4}\)
\(\Leftrightarrow x=\frac{3}{2}\).
