Use the definition of big-O to prove that 12 + 22 + ... + n2 is O(n3)
1^2+2^2+3^2+.....+n^2=((n*(n+1)*(2*n+1)))/6
=((n^2+n)*(2*n+1))/6
=(2*n^3+n^2+2*n^2+n)/6
=(2*n^3+3*n^2+n)/6
f(n)=(2*n^3+3*n^2+n)/6
g(n)=n^3
c=3
n>=n0 = 1
(2*n^3+3*n^2+n)/6 <= 3*n^3
So time complexity of 1^2+2^2+3^2+.....+n^2 is O(n^3)
please be clear with the steps taken and understandable 1. Prove that if f(n) = Θ(n2) for all f(n), then ΣΑ(n)-6(n3). i=1 2. Prove that if f.(n) are linear functions - i.e., that f(n)-Θ(n) for all Tn A(n) then Σ if.(n) = Θ(n3). Y definition of Big-Oh. ou are not required to use the formal i1
32 points Prove each of the following statements by applying the definition of Big-O. That is, derive an inequality (show your work) and identify the witness constants C and k as per the definition of Big-O.
-1) Prove that 12+22 + ... + n2 = n(n + 1) (2n + 1) 6 -1) Prove that 12+22 + ... + n2 = n(n + 1) (2n + 1) 6
Formal Definitions of Big-Oh, Big-Theta and Big-Omega: 1. Use the formal definition of Big-Oh to prove that if f(n) is a decreasing function, then f(n) = 0(1). A decreasing function is one in which f(x1) f(r2) if and only if xi 5 r2. You may assume that f(n) is positive evervwhere Hint: drawing a picture might make the proof for this problem more obvious 2. Use the formal definition of Big-Oh to prove that if f(n) = 0(g(n)) and g(n)...
Prove each of the following using the definition of Big-Oh. a)(?+1)5is O(?5) b)2?+1is O(2?) c)If ?(?)is a polynomial in ?, then ????(?)is ?(log?)
a) Prove that running time T(n)=n3+30n+1 is O(n3) [1 mark] b) Prove that running time T(n)=(n+30)(n+5) is O(n2) [1 mark] c) Count the number of primitive operation of algorithm unique1 on page 174 of textbook, give a big-Oh of this algorithm and prove it. [2 mark] d) Order the following function by asymptotic growth rate [2 mark] a. 4nlogn+2n b. 210 c. 3n+100logn d. n2+10n e. n3 f. nlogn
[12 marks] Using the definition of big-O, show that f(x) is big-O of g, where: f(x) = 2* + 33 and g(x) = 3* Show the details of your work to obtain a full mark.
Please explain big O. I don't get it Prove the following, using either the definition of Big-O or a limit argument. (a) log_2 (n) elementof O(n/log_2(n)) (b) 2^n elementof O(n!) (c) log_2(n^2) + log_2 (100n^10) elementof O(log_2 (n)) (d) n^1/2 elementof O(n^2/3) (e) log(3n) elementof O(log(2n)) (f) 2^n elementof O(3^n/n^2)
Prove the following using the following definition of O,Big-omega,Theta, small omega Σki=1 ?i ?i = ?(nk )??? ? > 1.
Prove (using the definition of O) or disprove (via counter-example): If f(n) = O(n)), and g(n) = O(n2), then f(n) + g(n) = O(n5). Prove (using the definition of O) or disprove (via counter-example): If f(n) = O(n), and g(n) = O(n2), then fin)/g(n) = O(n).