Let a_ i =i+ 1 i for i=1,2, ., 20 . Put p = 1 20 ( a ₁+ a ₂+ + a ₂₀ ) and q = 1 20 ( 1 a ₁ + 1 a ₂ + + 1 a ₂₀ ) . Then

Step-by-step solution

q >0 , try to the contra prove that q < 22- p 21 q + p 21 < 22 21 q + p 21 = 1 20 [ _ i =1 ²⁰ 1 a _ i + 1 21 _ i =1 ²⁰ a _ i ]= 1 20 [ _ i=1 ²⁰ ( i i ²+1 )+ 1 21 _ i =1 ²⁰ ( i + 1 i ) ]= 1 2 + 1 20 [ _ i =1 ²⁰ i i ²+1 + _ i =1 ²⁰ ( 1 21 i ) ]= 1 2 + 1 20 [ 1 2 + _ i =2 ²⁰ i i ²+1 + _ i =1 ²⁰ 1 21 i ] < 1 2 + 1 20 [ 1 2 + 2 5 _ i =2 ²⁰ 1+ 1 21 _ i =1 ²⁰ 1 ] < 1 2 + 1 20 [ 1 2 + 2 5 19+ 1 21 20 ] < 1 2 + 1 20 (1+8+1) < 1 2 + 1 2 < 22 21