If nЄN, find and prove a formula for...?

Question

(from i=1 to n)Σ (2i-1).

Answer 1

n
∑(2i - 1) = n^2
i=1

Let n = 1
2* 1 - 1 = 1

let n = 5
1 + 3 + 5 + 7 + 9 = 25

Let n = 6
1 + 3 + 5 + 7 + 9 + 11 = 36

Q.E.D.

Answer 2

There's a cute combinatorial proof too.

e e e e e
d d d d e
c c c d e
b b c d e
a b c d e

Notice that it's 1 a, 3 b's, 5 c's, 7 d's, 9 e's, making 25 total. Each successive odd number makes another "hook" for the next larger square.

(Some people don't consider this sort of reasoning to be as formal, but it does seem to help people at least understand in a different way than "symbol-pushing.")

Answer 3

This problem is often cited as the first example of the use of mathematical induction to prove a theorem about the natural numbers.

Answer 4

i^2. Prove by induction.