Question

S= 5/2 mũ 2 +5/3 mũ 2 + 5/4 mũ 2 + ....+5/100 mũ 2 chứng tỏ rằng 2<S<5

Answer 1

Ta có: \(\frac{1}{2^2}< \frac{1}{1.2}=\frac{1}{1}-\frac{1}{2}\); \(\frac{1}{3^2}< \frac{1}{2.3}=\frac{1}{2}-\frac{1}{3}\); ...; \(\frac{1}{100^2}< \frac{1}{99.100}=\frac{1}{99}-\frac{1}{100}\)

=> S < \(5\left(1-\frac{1}{100}\right)=5.\frac{99}{100}< 5.1=5\)=> S<5

Lại có: \(\frac{1}{2^2}>\frac{1}{2.3}=\frac{1}{2}-\frac{1}{3}\); \(\frac{1}{3^2}>\frac{1}{3.4}=\frac{1}{3}-\frac{1}{4}\); \(\frac{1}{100^2}>\frac{1}{100.101}=\frac{1}{100}-\frac{1}{101}\)

=> \(S>5\left(\frac{1}{2}-\frac{1}{101}\right)=5.\frac{101-2}{2.101}=\frac{5.99}{2.101}~2,45\)=> S>2

Vậy 2 < S < 5 => Đpcm