Please help! How can I prove that 2 to the power n is greater than 3n for all positive integers n is greater than or equal to 4?
You can use induction.
Is the statement true when n has its least value (i.e. when n = 4)? Yes, as 2^4 = 16 > 3 * 4 = 12
Now assume that the statement is true when n = k where k is any positive integer. So we assume that:
2^k > 3k Call this Equation 1
Now what would this assumption imply for 2^(k+1)?
2^(k+1) = 2 * 2^k > 3k (this last inequality is by equation 1)
So we have shown that IF 2^k > 3k then 2^(k+1) > 3k
Well, 2^k IS greater than 3k if k = 4 and so it must also be when k = 5,6,7.............