Please help me solve this question: Prove that 2^n > 3n for all positive integers n `>=` 4 Thanks...  

Expert Answers
jeew-m eNotes educator| Certified Educator


This can be solved by mathematical induction.

It has given that n>=4.

So the lowest value of n=4



`2^4 = 16`

`3xx4 = 12`


So when n = 4 the result is true.


Let us assume when n = p where p>4 the result is true.



Now we have to prove that when `n = p+1` the result is true.





What we have to show that is `2^(p+1)>3(p+1)` .

But what we have obtained is `2^(p+1)>6p` .

So if we can prove `6p>3(p+1)` then we have say;


So ultimately we will endup with `2^(p+1)>3(p+1)`


Let us assume `6p>3(p+1)`


`6p-3(p+1) > 0`


`3(p-1) > 0`

We know that `p>4` . Then `(p-1)>0` and `3(p-1) > 0` .

This means our assumption is correct.



Now we can say `2^(p+1)>3(p+1)`

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


Therefore from mathematical induction for all `n>=4` ;