Prove or disprove 42n omega 4n What is the growth of n2 2n

Prove or disprove: 4^2n = omega (4^n) What is the growth of n^2 + 2n^2 + 3n^2 + ... + n^4?

Solution

13 ) Since W isnt defined in the question I am going to assume that it is a constant and we need to prove that no matter what n is 4^2n can be written as w(4^n )

The exponent product rules tells us that the powers can be added if the base is same, we will manipulate the equation as

4^2n - w 4^n=0 , taking 4powerN common from the equation

we get ,

4^n[4^2-w]=0 , so either 4^n =0 or [4^2-w]

but  4^n will not be zero , for all n , so [4^2-w]=0 , 4^2=w

Hence 4^2n = w 4^n , put value of w , we get,

4^2n = 4^2 4^n , so it is disproved as 4^2n \\= 4^(2+n). Hpe this helps

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