# Visualising the sum of the first $n$ positive odd integers [duplicate]

Using the fact that $$1+2+\cdots+n=\frac{n(n+1)}{2}$$, we can deduce that sum of first $$n$$ positive odd integers is $$n^2$$. However, is there a way of finding the sum of $$1+3+5+\cdots+(2n-1)$$ visually?

• en.wikipedia.org/wiki/Proof_without_words#Sum_of_odd_numbers Aug 10 '20 at 14:34
• youtube.com/watch?v=IJ0EQCkJCTc Aug 10 '20 at 14:39
• math.stackexchange.com/a/733805/179297 Aug 10 '20 at 14:47
• $\displaystyle \begin{array}{ccc} \color{red}{\huge\bullet} & \color{magenta}{\huge\bullet} & \color{black}{\huge\bullet} \\ \color{magenta}{\huge\bullet} & \color{magenta}{\huge\bullet} & \color{black}{\huge\bullet} \\ \color{black}{\huge\bullet} & \color{black}{\huge\bullet} & \color{black}{\huge\bullet} \end{array}$ Aug 11 '20 at 5:19

## 1 Answer

Here is a ‘proof’ I once found in a book for young children. It is not a real proof in the mathematical sense, but rather a convincing example that any mathematician feels could be transformed into a rigourous proof:

Imagine wooden cubes stacked in rows, with the basis containing, say, $$7$$ cubes, the row above, $$5$$ cubes, the row still above, $$3$$ and the last row $$1$$, like this:

It is a geometrical evidence that, moving the grey squares from the bottom right corner to the top left corner, one recreates a square with sides equal to the number of rows, i.e. $$4$$ units, hence we have $$16$$ of them for the sum of the $$4$$ first odd numbers.