Math Problem..help with this equation?

Question

The maximum number of truck routes, t, that are needed to provide direct services to n cities, where no three cities lie in a stright line, given by the equation t= n^2 -- n all over / or divided by 2. How many truck routes will be needed to run service between 7 cities???

The answer I got was 21 truck routes. Is that correct?

Answer 1

The equation is (n² - n) / 2

For n = 7:

49 - 7
-------- = 21
.. 2

That is correct!

Note: Another way to write the formula is:
n(n - 1) / 2

This is sometimes easier/quicker to figure:
7 x 6
------- = 21
.. 2

The formula comes from thinking about each of the cities. With 7 cities, they have to each connect to 6 other cities. So 7 x 6 makes sense. But that double counts the routes, because a road between city A and city B is the same as a road between city B and city A. That's why you divide by 2.

Interestingly, the formula also works to figure out the sum of:
1 + 2 + 3 + 4 + 5 + 6 = 21

If you think about it... the first city connects to 6 others.
The next city connects to 5 others (you already counted the first city).
The next city connects to 4 others (you already counted the first 2).
etc.
The last city is already connected, so it is 0.

The formula for the sum of k numbers is:
[ (k + 1)k ] / 2

In your case k is one less than n, so let's substitute in n - 1
[ (n - 1 + 1)(n - 1) ] / 2
[ n (n - 1) ] / 2

I know this all more information than you asked, but I didn't just want to say, "yes, you are correct".

Answer 2

t = (7^2-7)/2 = 42/2 = 21
Right you are!!