work out f(n), the exact number of unit-time operations each procedure requires as a function of the input size n

Question

I have these 2 questions - I believe one was posted earlier - but was unanswered. I think I have answered 2nd one correctly ...

Any suggestions ?

work out f(n), the exact number of unit-time operations each procedure requires as a function of the input size n

(i) for i<-1 to n do
 for j<-1 to 2n-i do
//a unit cost operation
  ------------------------------ (i) this is i the one i need some help on.
(ii) for i <-1 to n do
 for j <- 2 to (n+i) do
// a unit cost operation

for i: 1 to n do:
 for j: 2 to n + i do:
  unit

now, let's say n=1

 i=1; j: 2 to 2 = 1 times
 total: 1 units

  n=2

i=1; j: 2 to 3 = 2 times
i=2; j: 2 to 4 = 3 times
 total: 2 + 3 = 5 units

 n=3

i=1; j: 2 to 4 = 4 times
i=2; j: 2 to 5 = 5 times
i=3; j: 2 to 6 = 6 times
total: 4 + 5 + 6 = 10 units

  Pattern being f(n) = n^2 +1 , is that right?

Answer 1

For the case you mentioned(case (i)) you can use Arithmetic Progressions

In this case as first term=2*n-1 and the last term is n, so sum of all the terms is

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

For the Case II, first term=(n+1-1)=n and last term=(n+n-1)=(2*n-1), so Sum, S is equal to,

S=n/2(first term+last term)=n/2*(n+2*n-1)=O(n^2)

You must have observed that S for both (i) & (ii) are same.