Question

Chứng minh rằng:

a ) 1 101 + 1 102 + 1 103 + ... + 1 150 > 1 3 b ) 1 201 + 1 202 + 1 203 + ... + 1 400 < 1 c ) 3 5 < 1 31 + 1 32 + 1 33 + ... + 1 60 < 4 5

Answer 1

a ) 1 101 + 1 102 + 1 103 + ... + 1 149 + 1 150 > 1 150 + 1 150 + ... + 1 150 ⏟ 50 s o = 1 3

b ) 1 201 + 1 202 + ... + 1 400 < 1 200 + 1 200 + ... + 1 200 ⏟ 200 s o = 1

c) đặt S= 1 31 + ... + 1 40 + 1 41 + ... + 1 50 + 1 51 + ... + 1 60

S > 1 40 + .. + 1 40 ⏟ 10 s o + 1 50 + ... + 1 50 ⏟ 10 s o + 1 60 + ... + 1 60 ⏟ 10 s o = 1 40 + 1 50 + 1 60 = 37 60

M à       37 60 > 36 60 = 3 5 = > S > 3 5

S < 1 30 + .. + 1 30 ⏟ 10 s o + 1 40 + ... + 1 40 ⏟ 10 s o + 1 50 + ... + 1 50 ⏟ 10 s o = 10 30 + 10 40 + 10 50 = 47 60

M à       47 60 < 48 60 = 4 5 = > S < 4 5