Home
Class 12
MATHS
Prove that (n !)^2<n^n(n !)<(2n)! for al...

Prove that `(n !)^2`<`n^n(n !)<(2n)!` for all positive integers n

Text Solution

Verified by Experts

We have,
`(n!)^(2)=(n!)=(1xx2xx3xx4xx.. Xx(n-1)n)(n!)`
Now, ` 1 le n, 2 le n, 3 le n, .., n le n`
or `1xx2xx3..(n-1)n le n xx n xx n..n`
or `n! le n^(n)`
or `(n!)(n!) le (n!)n^(n)`
or `(n!)^(2) le n^(n) (n!)`
Also, `(2n)!=1xx2..n xx (n+1)..(2n-1)xx(2n)`
Now, `n+1 gt n, n+2 gt n, n+3 gt n, .., n+n gt n`
or `(n+1)(n+2)(n+3)..(2n-1)(2n) gt n^(n)`
or ` n!(n+1)(n+2)..(2n-1)(2n) gt n!n^(n)`
or `(2n)! gt n! n^(n)`
`implies n! n^(n) lt (2n)!`
From (1) and (2), we get `(n!)^(2) le n^(n)(n!)lt (2n)!`
Promotional Banner

Similar Questions

Explore conceptually related problems

Prove that (n !+1) is not divisible by any natural number between 2 and n

Prove that (n !)! is divisible by (n !)^((n-1)!)

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

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

Prove that in S_(N)2 reaction inversion of configuration takes place.

If the ratio of the roots of the quadratic equation ax^(2)+bx+c=0 be m:n, then prove that ((m+n)^2)/(mn)=(b^(2))/(ac)

Prove that (3!)/(2(n+3))=sum_(r=0)^n(-1)^r((n C_r)/((r+3)C_3))

If c cos^(3)theta+3c cos theta sin^(2)theta=m and c sin^(3)theta+3c cos^(2)theta sin theta=n , then prove that (m+n)^((2)/(3))+(m-n)^((2)/(3))=2c^((2)/(3)).

Prove that, C_(0)^(2)+C_(1)^(2)+C_(2)^(2)+.......+C_(n)^(2)=((2n)!)/(n!)^(2)

For ninN , prove that ((n+1)/2)^ngt n!