prove by induction on n that for all integers n>=10, 2^n>n^3. you should not use

Question

prove by induction on n that for all integers n>=10, 2^n>n^3. you should not use calculus or any sophisticated mathematics, simple algebra is enough. But you should be precise enough that every step is clear (such as a simple algebra step) or that you explain/justify the step. Problem 1 (2 points). Prove by induction on n that for all integers n 10,2" > n You should not use calculus or any sophisticated mathematics, simple algebra is enough. But you should be precise enough that every step is clear (such as a simple algebra step) or that you explain/justify the step.

Explanation / Answer

For n>=10

2^n > n^3

Let us prove it using mathematical induction

1) For n=10

2^10 > 10^3

1024 > 1000

2) For n=k

Assume it is true for n=k which means

2^k > k^3

3) Let us try to prove that it is true for n=k+1

It can be proved if we arrive at the conclusion 2^k+1 > (k+1)^3

The above is equivalent to-

2^(k+1) > (k+1)^3

k+1 < 2^(k+1)/3

RHS-

2^0.33 * 2^(k/3)

We have 2^0.33 * 2^(k/3) > 2^0.33 * k and  

k*2^(k/3) > k+1

k(2^(k/3)-1)>1

k>1/(2^(k/3)-1)

This is true for k>=4

which means (k+1)^3 < 2^k+1

Hence proved by induction

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