Understanding the Statement 2^N < N^4 Using Proof by Induction
The inequality (2^N < N^4) proposes that for all integers (N) greater than a certain threshold, (2^N) grows at a slower rate than (N^4). The goal is to establish this relationship through mathematical induction, a powerful proof technique frequently utilized in number theory and combinatorics.
Base Case
To initiate a proof by induction, it is essential to verify the base case. The base case typically involves checking the smallest integer for which the statement is presumed to hold. For the inequality (2^N < N^4), the smallest integer (N) we suspect might hold true is (N = 5):
- For (N = 5):
- Calculate (2^5 = 32)
- Calculate (5^4 = 625)
- Clearly, (32 < 625)
This indicates that the inequality holds for (N = 5), providing a valid starting point for our induction process.
Inductive Step
After establishing the base case, the next step involves assuming the statement is true for some arbitrary integer (k), where (k \geq 5). This assumption is called the inductive hypothesis:
- Assumption: (2^k < k^4)
The goal is to prove that this assumption implies the inequality also holds for the next integer, (k + 1):
- Show that (2^{k+1} < (k + 1)^4)
Starting from the left side of the inequality, we can rewrite (2^{k+1}):
[2^{k+1} = 2 \cdot 2^k
]
Using the inductive hypothesis, substitute (2^k):
[2^{k+1} < 2 \cdot k^4
]
Next, it is necessary to demonstrate that:
[2 \cdot k^4 < (k + 1)^4
]
Expanding ((k + 1)^4) using the Binomial Theorem:
[(k + 1)^4 = k^4 + 4k^3 + 6k^2 + 4k + 1
]
To verify the inequality (2 \cdot k^4 < k^4 + 4k^3 + 6k^2 + 4k + 1), we will simplify both sides:
[2k^4 < k^4 + 4k^3 + 6k^2 + 4k + 1
]
This can be rewritten as:
[k^4 < 4k^3 + 6k^2 + 4k + 1
]
Analyzing the Inequality
Now, we can examine this inequality. Factoring out (k^4) on the left implies that we need to check whether this holds for (k \geq 5).
Testing for (k = 5):
[5^4 < 4 \cdot 5^3 + 6 \cdot 5^2 + 4 \cdot 5 + 1
]
Calculating each side:
- Left side: (625)
- Right side: (4 \cdot 125 + 6 \cdot 25 + 20 + 1 = 500 + 150 + 20 + 1 = 671)
The inequality (625 < 671) holds true. For increasing (k), it can be observed that both the left side and right side of our original inductive inequality rise, but the right side grows significantly faster due to the cubic (4k^3) term.
Completion of Inductive Step and Proof
Therefore, since the assumption (2^k < k^4) implies (2^{k+1} < (k + 1)^4), we can conclude that by mathematical induction the inequality (2^N < N^4) holds for all integers (N \geq 5).
FAQ
What is mathematical induction?
Mathematical induction is a method of mathematical proof typically used to establish that a given statement holds for all natural numbers. It involves two steps: proving the base case and proving the inductive step.
Why does the inequality (2^N < N^4) only hold for (N \geq 5)?
The growth rates of the functions (2^N) and (N^4) vary significantly for small values of (N). For (N < 5), (2^N) can be greater than or equal to (N^4), but starting from (N = 5), (N^4) consistently outpaces (2^N).
Can this proof technique be applied to other inequalities?
Yes, mathematical induction is a versatile proof technique that can be applied to various types of inequalities and properties in mathematics, particularly those involving natural numbers.