Home
Class 12
MATHS
Prove that ((n + 1)/(2))^(n) gt n!...

Prove that `((n + 1)/(2))^(n) gt n!`

Text Solution

Verified by Experts

We have
`(1+2+3+4+....+n)/(n) gt (1.2.3.4....n)^(1//n)`
`(m(n+1))/(2))/(n) gt (n!)^(1//n)`
`rArr ((n+1)/(2))^n gt n!`
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • INEQUALITIES INVOLVING MEANS

    CENGAGE|Exercise Exercise 6.3|6 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE|Exercise Exercise 6.4|4 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE|Exercise Exercise 6.1|8 Videos

  • INEQUALITIES AND MODULUS

    CENGAGE|Exercise Single correct Answer|21 Videos

  • INTEGRALS

    CENGAGE|Exercise Solved Examples And Exercises|324 Videos

Similar Questions

Explore conceptually related problems

Prove that 3^(n+1)gt3(n+1)

Using the principle of mathematical induction. Prove that (1^(2)+2^(2)+…+n^(2)) gt n^(3)/3 " for all values of " n in N .

Prove that ((2n)!)/(2^(2n)(n!)^(2))<=(1)/(sqrt(3n+1)) for all n in N

Prove that (.^(2n)C_(0))^(2) - (.^(2n)C_(1))^(2) + (.^(2n)C_(2))^(2) - …. + (.^(2n)C_(2n))^(2) = (-1)^(n) .^(2n)C_(n)

Prove that [n+1/2]^(n)>(n!)

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 by induction that : 2^(n) gt n for all n in N .

Prove that: n!(n+2)=n!+(n+1)!

Prove that ((2n+1)!)/(n!)=2^(n)[1.3.5.....(2n-1)*(2n+1)]

Prove that ((2n+1)!)/(n!)=2^(n){1.3.5(2n-1)(2n+1)}

Home
Profile