Respuesta :

josebarradas1607

Step-by-step explanation:

The statement to be proved using mathematical induction is:

  1. "For every [tex]n\geq 4[/tex], [tex]5^n\geq 2^{2n+1}+100[/tex]

We will begin the proof showing that the base case is satisfied (n=4).

[tex]5^4=625\geq 612=2^{2*4+1}+100[/tex].

Then, 1 is true for n=4.

Now we will assume that the statement holds for some arbitrary natural number [tex]n\geq 4[/tex] and prove that then, the statement holds for n+1. Observe that

[tex]2^{2(n+1)+1}+100=2^{2n+1+2}+100=4*2^{2n+1}+100\leq 4(2^{2n+1}+100)\leq 4*5^n<5^{n+1}[/tex]

With this the inductive step has been proven and then, our statement is true,

For every [tex]n\geq 4[/tex], [tex]5^n\geq 2^{2n+1}+100[/tex]

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