Power by using the principle of Mathematical Induction that `n!>= 2^(n-1)` for every positive integer n.

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jeew-m eNotes educator| Certified Educator

`n! >= 2^(n-1) `

`n = 1`

`LHS = 1! =1`

`RHS = 2^(1-1) = 1`


For `n = 1` result is true.

Let us assume that for n = p where `p>=1` and a positive integer the result is true.


`n = p+1`

`p!>=2^(p-1)` multiply both sides by `(p+1` ).


`(p+1)!>= 2^(p-1)xx(p+1)`

`p>=1` therefore `(p+1)>=2`


`(p+1)!>= 2^(p-1)xx2` is valid

`(p+1)!>= 2^(p+1-1)`

So for n = p+1 the result is true.

So from mathematical induction it is proved that `n! >= 2^(n-1) `