ok so I have to create an equation to explain how many handshakes you get if you input any number of people. for instance, i came up with this equation:
n=number of people
n(n-1)
-------
2

so we know it would take 15 handshakes for 6 people to all shake everyones hand...and 21 for 7 people, and 28 for 8 people and 36 for 9 people and so on. so when you put 7 in as n, you get 42, then you divide by 2 and get 21, which is the number of handshakes.

but the other part of the problem is that the equation has to be manipulated to be true for (x+1)

therefore; (x+1)((x+1)-1)
-----------------
2

kind of like that. except for that then it isn't true. so i have to manipulate an equation to be true for x=1 and x=x+1
which is causing a serious problem
and i have to show how i manipulated it to get there.
so if anyone can help, please do!!
thanks!

2. you are using induction to prove that the number of handshakes is

$h = \frac{n(n-1)}{2} = \frac{n^2}{2} - \frac{n}{2}$

you assume it's true for $n$, then prove true for $n+1$

if you add one more person, then he/she shakes hands with n people. making the total number of handshakes $h + n$

so ... for $n+1$ people

$\frac{(n+1)(n)}{2} = \frac{n^2}{2} + \frac{n}{2} = \frac{n^2}{2} + n - \frac{n}{2} = \frac{n^2}{2} - \frac{n}{2} + n = h + n$