Respuesta :
Answer:
The inequality that you have is [tex]5^{n}>2^{2n+1}+100,\,n>4[/tex]. You can use mathematical induction as follows:
Step-by-step explanation:
For [tex]n=5[/tex] we have:
[tex]5^{5}=3125[/tex]
[tex]2^{(2(5)+1)}+100=2148[/tex]
Hence, we have that [tex]5^{5}>2^{(2(5)+1)}+100.[/tex]
Now suppose that the inequality holds for [tex]n=k[/tex] and let's proof that the same holds for [tex]n=k+1[/tex]. In fact,
[tex]5^{k+1}=5^{k}\cdot 5>(2^{2k+1}+100)\cdot 5.[/tex]
Where the last inequality holds by the induction hypothesis.Then,
[tex]5^{k+1}>(2^{2k+1}+100)\cdot (4+1)[/tex]
[tex]5^{k+1}>2^{2k+1}\cdot 4+100\cdot 4+2^{2k+1}+100[/tex]
[tex]5^{k+1}>2^{2k+3}+100\cdot 4[/tex]
[tex]5^{k+1}>2^{2(k+1)+1}+100[/tex]
Then, the inequality is True whenever [tex]n>4[/tex].
