3 squared can be expressed as...?

Question

3^2 = 2 x 3+ 2 x 1+1
Using this expression, how could n^2 be expressed?

So far I have n^2=(n - 1) x n + (n-1) x (n-2) + (n-2) but I don't think it is correct. Help please?

Answer 1

If you take your example, notice that 2 is repeated. So let's factor it out:

3² = 2(3 + 1) + 1
3² = 2(4) + 1

This means that:
n² = (n - 1)(n + 1) + 1

How can we get there?

Let's look at (k - 1)²

That expands to:
(k - 1)² = k² - 2k + 1

(k - 1)² = (k - 2)k + 1

Now replace k with n + 1:
n² = (n - 1)(n + 1) + 1

Alternatively, this can be expanded back out to:
n² = (n - 1)n + (n - 1) + 1

To match your example:
3² = 2 x 3 + 2 + 1

Let's try it for another number, like 11²
11² = 10 x 11 + 10 + 1
11² = 110 + 10 + 1
11² = 121

Or 5²
5² = 4 x 5 + 4 + 1
5² = 20 + 4 + 1
5² = 25

Seems to work...

Answer 2

Your formula is completely wrong.

3-squared is 3 x 3 which is equal to 9, yes, but 4-squared, by your formula becomes 20, not 16. And the difference gets worse as "n" increases

Puzzling's formula is a trivialized equivalency, since he's doing nothing (by the (n-1)(n+1) factoring) more than subtracting out "1" from the number squared, and then adding it back in. Obviously it is the same as n-squared. His throwing in the K-1 value is a red herring, adds complexity and makes it look like there is some complex mathematics at work.

But his formula is not the same as yours, anyway.

Answer 3

n^2 = 1+3+5 +... +2n-1

Answer 4

It sure isn't.

I would go for
2x + (x - 1) (x - 2) + (x-2)

Answer 5

Are you looking for a template or pattern?