hình như cậu ghi sai đề?

\(A=\left(\frac{1}{2^2}-1\right).\left(\frac{1}{3^2}-1\right).......\left(\frac{1}{100^2}-1\right)\)

\(A=\left(\frac{1}{2^2}-\frac{2^2}{2^2}\right).\left(\frac{1}{3^2}-\frac{3^2}{3^2}\right).....\left(\frac{1}{100^2}-\frac{100^2}{100^2}\right)\)

\(A=\left(-\frac{3}{4}\right).\left(-\frac{8}{9}\right)........\left(-\frac{9999}{10000}\right)\)

\(A=\frac{\left(-3\right).\left(-8\right).....\left(-9999\right)}{4.9...10000}=\frac{1.\left(-3\right).2.\left(-4\right)......99.\left(-101\right)}{2.2.3.3.....100.100}\)

\(A=\frac{\left(1.2.3....99\right).\left[\left(-3\right).\left(-4\right)......\left(-101\right)\right]}{\left(2.3.4....100\right).\left(2.3.4...100\right)}=\frac{1.\left(-101\right)}{100.\left(-1.\right).\left(-1\right)....\left(-1\right).2}=\frac{-101}{100.2}=\frac{-101}{200}\)

Ta thấy \(\frac{-101}{200}< \frac{-100}{200}=\frac{-1}{2}\Rightarrow A< -\frac{1}{2}\)

\(A=\frac{-3}{4}.\frac{-8}{9}......\frac{-9999}{1000}\)

\(=-\frac{1.3}{2.2}.\frac{2.4}{3.3}....\frac{99.101}{100.100}\)

\(=-\frac{1.2.3...99}{2.3...100}.\frac{3.4...101}{2.3...100}\)

\(=-\frac{1}{100}.\frac{101}{2}=-\frac{101}{200}< \frac{-100}{200}=\frac{-1}{2}\)

VẬY \(A< \frac{-1}{2}\)

Đổi: 675km = 67 500 000cm

Trên bản đồ tỉ lệ 1:2 500 000 quãng đường dài là:

67 500 000 : 2 500 000 = 27 (cm)

Đáp số: 27 cm 

Xin lỗi nha

\(A=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2016}}\)

Ta thấy \(A=1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^{2015}}=1+A-\frac{1}{2^{2016}}\)

\(\Rightarrow A=1-\frac{1}{2^{2016}}< 1\)

Vậy A < 1.