Question

 cho S=\(\frac{3}{1\cdot4}\)+\(\frac{3}{4\cdot7}\)+\(\frac{3}{7\cdot10}\)+.....+\(\frac{3}{40\cdot43}\)+\(\frac{3}{43\cdot46}\).hãy chứng tỏ rằng S nhỏ hơn 1

Answer 1

\(^{\frac{3}{1\cdot4}+\frac{3}{4\cdot7}+\frac{3}{7\cdot10}+...+\frac{3}{40\cdot43}+\frac{3}{43\cdot46}}\)

\(1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{10}-\frac{1}{43}+\frac{1}{43}-\frac{1}{46}\)

\(1-\frac{1}{46}=\frac{45}{46}\)

Vì \(1-\frac{1}{46}< 1\)nên \(\frac{3}{1\cdot4}+\frac{3}{4\cdot7}+\frac{3}{7\cdot10}+...+\frac{3}{40\cdot43}+\frac{3}{43\cdot46}< 1\)

Chúc bạn học tốt

Answer 2

\(S=\frac{3}{1\cdot4}+\frac{3}{4\cdot7}+\frac{3}{7\cdot10}+...+\frac{3}{40\cdot43}\)

\(S=1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+\frac{1}{7}-\frac{1}{10}+...+\frac{1}{40}-\frac{1}{43}\)

\(S=1-\frac{1}{43}\)

\(S=\frac{42}{43}< 1\)