Respuesta :
Answer:
The given expression is divisible by 3 for all natural values of x.
Step-by-step explanation:
The given expression is
[tex]2^{2x+1}+1[/tex]
For x=1,
[tex]2^{2(1)+1}+1=2^{3}+18+1=9[/tex]
9 is divisible by 3. So, the given statement is true for x=1.
Assumed that the given statement is true for n=k.
[tex]2^{2k+1}+1[/tex]
This expression is divisible by 3. So,
[tex]2^{2k+1}+1=3n[/tex] .... (1)
For x=k+1
[tex]2^{2(k+1)+1}+1[/tex]
[tex]2^{2k+2+1}+1[/tex]
[tex]2^{(2k+1)+2}+1[/tex]
[tex]2^{2k+1}2^2+1[/tex]
Using equation (1), we get
[tex](3n-1)2^2+1[/tex]
[tex](3n)2^2-2^2+1[/tex]
[tex](3n)2^2-4+1[/tex]
[tex](3n)4-3[/tex]
[tex]3(4n-1)[/tex]
This expression is also divisible by 3.
Therefore the given expression is divisible by 3 for all natural values of x.
