Question

Aratati ca:
a)[tex] \frac{2^{n+2}*5^{n+2}+2^{n}*5^{n+3}}{4*2^{n}*5^{n+1}+2^{n}*5^{n}*25} apartine~N
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b) [tex] \frac{3^{n+2}*5^{n}+15^{n+1}+3^{n}*5^{n}}{3^{n+3}*5^{n}-2*3^{n}*5^{n}} apartine~N

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Rezolvare completa va rog!

Answer 1

[2^(n+2) × 5^(n+2) +2^n × 5^(n+3)] /[4×2^n × 5^(n+1) +2^n × 5^n ×25)]=
=[2^(n+2) × 5^(n+2) +2^n × 5^(n+3)] /[2^(n+2) × 5^(n+1) +2^n × 5^(n+2)]=
=[2^n × 5^(n+1) (2^2×5+5²)] /[2^n × 5^(n+1)(2^2 + 5)]=
=(20+25)/(4+5)=45/9=5


[3^(n+2) × 5^n +15^(n+1)+3^n × 5^n] /[3^(n+3) × 5^n -2×3^n × 5^n)]=
=[3^(n+2) × 5^n +3^(n+1)×5^(n+1)+3^n × 5^n] /[3^(n+3) × 5^n -2×3^n × 5^n)]=
=[3^n × 5^n(3^2 +3^1×5^1+1)] /[3^n × 5^n(3^3- 2)]=
=(9 +15+1)] /(27- 2)=
=25/25=1