Question

Problem 3.2.14a Show that 2^2x+1 +1 is divisible by 3.

Answer 1

Answer:

The given expression is divisible by 3 for all natural values of x.

Step-by-step explanation:

The given expression is

$2^{2x+1}+1$

For x=1,

$2^{2(1)+1}+1=2^{3}+18+1=9$

9 is divisible by 3. So, the given statement is true for x=1.

Assumed that the given statement is true for n=k.

$2^{2k+1}+1$

This expression is divisible by 3. So,

$2^{2k+1}+1=3n$              .... (1)

For x=k+1

$2^{2(k+1)+1}+1$

$2^{2k+2+1}+1$

$2^{(2k+1)+2}+1$

$2^{2k+1}2^2+1$

Using equation (1), we get

$(3n-1)2^2+1$

$(3n)2^2-2^2+1$

$(3n)2^2-4+1$

$(3n)4-3$

$3(4n-1)$

This expression is also divisible by 3.

Therefore the given expression is divisible by 3 for all natural values of x.