Question

[Kudo Shinichi]

Cho \(B=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{18.19.20}.\)

Chứng minh rằng:\(B< \frac{1}{4}.\)

Answer 1

[♨❤♨~Koo ๖ۣۜϑũ❥꧁şɦυυ꧂❣ ♫✔♫]

2B=\(\frac{2}{1.2.3}\)+.....+\(\frac{2}{18.19.20}\)

2B=\(\frac{1}{1.2}\)-\(\frac{1}{2.3}\)+\(\frac{1}{2.3}\)-\(\frac{1}{3.4}\).......+\(\frac{1}{18.19}\)-\(\frac{1}{19.20}\)

2B=\(\frac{1}{1.2}\)-\(\frac{1}{19.20}\)

B=\(\frac{1}{1.2}\):2-\(\frac{1}{19.20}\):2

B=\(\frac{1}{1.2}\).\(\frac{1}{2}\)-\(\frac{1}{19.20}\).\(\frac{1}{2}\)

=\(\frac{1}{4}\)-\(\frac{1}{19.20.2}\)<\(\frac{1}{4}\)

Answer 2

[Jenny Dolly Marion_ Love For You]

\(B=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{18.19.20}\)

\(2B=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{18.19}-\frac{1}{19.20}\)

\(2B=\frac{1}{1.2}-\frac{1}{19.20}\)

\(B=\left(\frac{1}{2}-\frac{1}{19.20}\right):2\)

\(B=\frac{189}{760}\)