I need help with a geometry question!!! please HELP!?

Question

Ok I am supposed to find a general expression about the relationship between "n" and "number of segments"

n= the number of points on a line.

if n=2 then you have 1 different segment

if n=3 then you have 3 different segments

if n=4 then you have 6 different segments

if n=5 then you have 10 different segments

if n=6 then you have 15 different segments

if n=7 then you have 21 defferent segments

and so on.

now I need to find a general expression that would give me the relationship between 'n' and 'number of segments'

that way you could just pick a random number, put it into the equation, and get the answer and not have to list out all the previous problems.

please help!!!

thank you.

Answer 1

If you notice, the answer is 1 + 2 + 3 + 4 + ... + (n-1)

The sum of the numbers 1 through (n-1) can be expressed using the following formula:

f(x) = n(n-1)/2

For example:

n = 6

f(6) = (6 * 5) / 2 = 15

Answer 2

Well, the key to this problem is that you must have two points to make a segment.  So if have a set of random points, say 10 of them, then i can use any two of the 10 to make a segment.  Well this takes us to counting practices.

We have n points and we will choose any 2 of them.

We will use the equation:

number of segments = nCr = nC2 = n!/((2)(n-2)! = n(n-1)/2

So we get:

number of segments = n(n-1)/2

Voila

Answer 3

n = 2, you have 0 + 1 segement

n = 3, you have 0 + 1 + 2 segments

n = 4, you have  0 + 1 + 2 + 3 segments

n = 5, you have 0 + 1 + 2 + 3 + 4 segments

and so forth, so...

if I have n ponits on a line, you'll have 0 + 1 + 2 + ... + (n -1) different segments, or...

(n/2)(n -1)  = number of different segments

for ex. if n = 8

(8/2)(8 - 1) = 28 different segments