Playing with squares

Question

Extending from particular examples I've found that $$n^2=\sum_{i=1}^{i=n-1} 2\, i+n$$ this is that for any square of side $n$ the area can be calculated in a simple way.

Example

For a square of side $7$, the result is: $2×1+2×2+2×3+\cdots + 7=49$

Question

Is there any way to prove this generally true, is there more than one way?Can you show at least one?

Answer 1

A visual proof. A variant of the one found in the book "Proofs Without Words"

Answer 2

Permuting your equation, i.e. moving $n$ to the left, and dividing by $2$, yields the equivalent equation

$$\frac{n(n-1)}{2}=\sum_{i=1}^{n-1} i$$ which is a very well known equation that can easily be proven by induction. Or just look at how Gauss did it when he was a child.

Answer 3

$$\sum_{i=1}^{i=n-1} 2\, i\,+n=2\left(\dfrac {n (n-1)}2\right)+n=n^2$$

Answer 4

Note that the sum can be written as a telescopic sum: $$\begin{align}\sum_{i=1}^{n-1} 2 i+n&=\sum_{i=1}^{n-1} (2 i+1)+1=\sum_{i=1}^{n-1} ((i+1)^2-i^2)+1\\ &=(n^2-1)+1=n^2. \end{align}$$ Similarly, we can show $$\sum_{i=1}^{n-1} (3i^2+3i)+n=n^3\quad\text{or}\quad \sum_{i=1}^{n-1} (4i^3+6i^2+4i)+n=n^4. $$