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#1 2006-08-14 04:03:04

nghoihin1
Guest

Maths problem

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?

#2 2006-08-15 03:51:24

gnitsuk
Member
Registered: 2006-02-09
Posts: 118

Re: Maths problem

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.............

Mitch.

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