we have to resolve the problem by induction proofabstract ma

we have to resolve the problem by induction proof(abstract math) For all integers n that are greater than 5, we have3^n<n!. Thanks

Solution

The simplest and most common form of mathematical induction infers that a statement involving a natural number n holds for all values of n. The proof consists of two steps:

The hypothesis in the inductive step that the statement holds for some n is called the induction hypothesis (or inductive hypothesis). To perform the inductive step, one assumes the induction hypothesis and then uses this assumption to prove the statement for n + 1.

First of all ,this is true for all integers n that are greater than 6 , (check for n = 6)

now for n= 7

3⁷ = 2187 7! = 5040 so we have 3^n<n!.   

now let us assume it it true for k

3^k < k! ... (!)

for (k+1) ,

3 ^(k+1)    = 3 * 3^k   < 3 * k! { using equation 1}

< (k+1) *k! { as k> 7 > 3 , so 3 * k! <  (k+1) *k! }

so it is true for ( k +1) too

hence proved .