Does thetan3 2n 1thetan3 holdSolutionTheta notation fn is

Does theta(n^3 + 2n + 1)=theta(n^3) hold?

Solution

Theta notation: -f(n) is theta of g(n)...iff there exits positive real constants k1 and k2 and positive integer n0, such that

k1*g(n)<=f(n)<=k2*g(n) for all n>n0..

now f(n)= n3+2n+1

g(n)= n3

if we can find k1,k2 and n0, such that

k1*n^3 <= n^3+2n+1 <= k2*n^3, n>n0

from above equation..

k1 =1, k2= 2, n0=1

1*n^3 <= n^3+2n+1<=k2*n^3,n>1

since, such constants k1,k2 and n0, that satisfy the equation, are exists...

hence

theata(n^3+2n+1) = theata(n^3).. both are equal hence proved..