Question

Chưng minh rằng:1/3^2+1/4^2+1/5^2+/6^2+....+1/100^2<1/2

Answer 1

Ta có: 1/3^2=1/3.3<1/2.3

            1/4^2=1/4.4<1/3.4

            1/5^2=1/5.5<1/4.5

            1/6^2=1/6.6<1/5.6

            ...............................

           1/100^2=1/100.100<1/99.100

=>1/3^2+1/4^2+1/5^2+1/6^2+....+1/100^2<1/2.3+1/3.4+1/4.5+1/5.6+....+1/99.100

=>1/3^2+1/4^2+1/5^2+1/6^2+....+1/100^2<1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+....+1/99-1/100

=>1/3^2+1/4^2+1/5^2+1/6^2+....+1/100^2<1/2-1/100

=>1/3^2+1/4^2+1/5^2+1/6^2+....+1/100^2<49/100      (1)

Ta có: 1/2=50/100>49/100        (2)

Từ (1) và (2) =>1/3^2+1/4^2+1/5^2+1/6^2+....+1/100^2<1/2(đpcm)