True / False For every intger `n >1` , the inequality `(n !)^(1//n)<(n+1)/2` holds

The inequality n!>2^(n-1) is true

Prove that for every positive integer n the following inequality holds true: (1)/(9)+(1)/(25)+…..+(1)/((2n+1)^2)lt(1)/(4) .

The number of positive integers satisfying the inequality C(n+1,n-2)-C(n+1,n-1)<=100 is

If (n+3)^(2)=a(n+2)^(2)+b(n+1)^(2)+cn^(2) holds true for every positive integer n, then the quadratic equation ax^(2)+bx+c=0 has

The number of positive integers satisfying the inequality ""^(n+1)C_(3) - ""^(n-1) C_(2) le 100 is

The least positive integer value of n satisfying the inequality C(10,n-1)>2C(10,n) is

For every integer n>=1,(3^(2^(n))-1) is divisible by

Let a gt 1 and n in N . Prove that inequality , a^(n)-1 ge n (a^((n+1)/(2))-a^((n-1)/(2))) .

Prove that for every positive integer n, 1^(n) + 8^(n) - 3^(n) - 6^(n) is divisible by 10.