Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

You are not logged in.

- Topics: Active | Unanswered

Pages: **1**

**nghoihin1****Guest**

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?

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

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.

Offline

Pages: **1**