Conjecture and telescoping sum! Can anybody explain these to me???

[SilverSprite]

Anybody ever heard of conjecture and/or telescoping sum? If so can you guys explain what it is?

[sql_lall]

A telescoping sum is one that you can 'cancel' each part of the sum to leave you with a very easy sum.

An example of a telescoping sum:

First:
f(x) = x² - (x-1)²

Now to find N = f(1) + f(2) + f(3) + ... + f(10), use a telescoping sum:
N=
1² - 0²
+ 2² - 1²
+ 3² - 2²
+....
+ 10² - 9²

= 10² - 0² = 100

(see how the + 2² and the - 2² cancel out)

[Kalkewl8ter]

Conjecture:
You consider small cases, and hypothesize a general formula.