Can someone explain to me how they got these formulas ?

Tutor: None Selected Time limit: 1 Day

You pay $56 to rent a hotel room, but if you rent more than one room, you get a discount of $2 per room for each additional room rented.

(c) Say you rent n rooms. Find a formula showing how much each one costs as a function of n.
R = (No Response) 

(d) Say you rent n rooms. Find a formula showing how much you will pay for all n rooms combined as a function of n.
C = (No Response) 

Oct 3rd, 2015


The first room costs 56, the second 56-2, the third 56-2-2. The general rule is simple:

k-th room costs 56-2*(k-1) which is equal to 58-2*k.

So if I understand correctly, the answer for (c) is 58-2*n.

Now let's sum these values for k=1..n:

Sum_k=1_k=n_[58-2*k] =
Sum_k=1_k=n_[58] - 2*Sum_k=1_k=n_[k].

The first Sum is 58*n and the second is n*(n+1)/2 (the sum of the arithmetic progression).

So the result is 58*n - n*(n+1) which is equal to 57*n-n^2 or n*(57-n). It is the answer for the (d).

Please ask if something is unclear.
Sep 15th, 2015

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