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Problem 3.2.14a

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

User Chrs
by
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1 Answer

4 votes

Answer:

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

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.

User Aky
by
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