Question

Hello, I need help with the following Discrete problem. Please show your work, thank you!

12. Select a theta notation from among (1), (n), (n?), (nº), (n), (2"), or (n!) for each of the expressions (a) 4n + 2n? -5 (b) 13 + 2 +...+ n (c) Prove you are correct for both parts a and b above

Answer 1

`Hey,

Note: If you have any queries related the answer please do comment. I would be very happy to resolve all your queries.

a)

Let f(n)=4*n^3+2*n^2-5

So,

f(n)=theta(n^3)

To prove it we will use limit theorem

Let g(n)=n^3

So, by limit theorem

f(n)=theta(g(n)) iff f/g=constant for n tending inf

So,

f/g=(4*n^3+2*n^2-5)/n^3=4+2/n-5/n^3=4=constant

So,

f(n)=theta(g(n))

Hence proved

b)

By euler formula

S=(n*(n+1)/2)^2

So,

Let f(n)=(n*(n+1)/2)^2

Let g(n)=n^4

To prove it we will use limit theorem

Let g(n)=n^3

So, by limit theorem

f(n)=theta(g(n)) iff f/g=constant for n tending inf

So,

f/g=(n*(n+1)/2)^2/n^4=(1/4)*(1+1/n)^2=1/4=constant

So,

f(n)=theta(n^4)

f(n)=theta(g(n))

Hence proved

Kindly revert for any queries

Thanks.