Problem 3.2.14aShow that 2^2x+1 +1 is divisible by 3.
Accepted Solution
A:
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.