Visualising number sequences

Numbers are easier to understand when we see them. A pile of $9$ pebbles is just a pile, but the same $9$ pebbles arranged as a $3 \times 3$ square suddenly tells us $9$ is a square number. Pictures unlock the secrets that long columns of figures hide.

Concept

Take the counting numbers $1, 2, 3, 4, 5, \dots$. If we draw each as a column of dots , one dot, then two, then three , we get a "staircase":

*
* *
* * *
* * * *
* * * * *

Counting the dots row by row gives us $1, 1+2, 1+2+3, 1+2+3+4, 1+2+3+4+5$, which is exactly the triangular sequence $1, 3, 6, 10, 15$. The staircase is the triangular numbers.

Now do something clever. Take two copies of the same staircase and turn one of them upside down. Fit them together. You get a rectangle that is $n$ tall and $n+1$ wide , exactly $n(n+1)$ dots. Since two triangles made the rectangle, one triangle has

$T_n = \frac{n(n+1)}{2}$

dots. This is one of the most famous formulas in school mathematics, and we discovered it just by looking at a picture.

For square numbers, the picture is even simpler: a $1 \times 1$ block, then a $2 \times 2$, then $3 \times 3$, then $4 \times 4$. Each new square is built by adding an L-shaped border called a gnomon to the previous one. The L-shape for the $n$th square has $2n - 1$ dots , exactly the next odd number. That is the visual reason adding odd numbers gives squares: each odd number is one more L-shape stacked on the square.

$1 + 3 + 5 + 7 + \dots + (2n-1) = n^2$

You can also visualise powers of 2. Draw one dot. Double it. Double again. Each step doubles the picture , like a tree that branches in two every level. By the tenth doubling you already have $1024$ dots.

Drawing sequences is not just pretty. It often gives the cleanest explanation for a rule. Pictures cannot be argued with: once you see two staircases making a rectangle, you know the triangular formula is true.

Worked examples

Example 1. Draw the first four triangular numbers as triangles of dots. Count the dots.

• $T_1 = 1$ dot.
• $T_2 = 1 + 2 = 3$ dots.
• $T_3 = 1 + 2 + 3 = 6$ dots.
• $T_4 = 1 + 2 + 3 + 4 = 10$ dots.

Example 2. Use the gnomon idea to find $1 + 3 + 5 + 7 + 9 + 11$.

• That is six odd numbers, so the answer is $6^2 = 36$.

Example 3. Use the formula $T_n = \frac{n(n+1)}{2}$ to find the $20$th triangular number.

• $T_{20} = \frac{20 \times 21}{2} = \frac{420}{2} = 210$.

Example 4. Two staircases of height $7$ are fit together to make a rectangle. What size is the rectangle? How many dots?

• Height $= 7$, width $= 7 + 1 = 8$.
• Dots $= 7 \times 8 = 56$.
• One staircase has $56 \div 2 = 28$ dots, matching $T_7$.

Try it yourself

1. Draw $T_5$ as a triangle. How many dots?
2. Draw the squares $1^2, 2^2, 3^2, 4^2$ and mark the L-shaped gnomons in colour.
3. Use the formula to find $T_{50}$.
4. Use the gnomon idea to find $1 + 3 + 5 + \dots + 19$.
5. Sketch the doubling sequence $1, 2, 4, 8, 16$ as growing groups of dots.
6. The first $10$ counting numbers sum to which triangular number?
7. Tricky: which triangular number is exactly $100$? Are there none, one, or more?

Activity

Two-triangle rectangle. Draw $T_6$ on paper , six rows of dots, $1$ then $2$ then $\dots$ then $6$. Cut it out. Make a second identical copy, flip it upside-down, and tape the two together along the slanted edge. You should get a $6 \times 7$ rectangle. Count the dots in the rectangle, divide by $2$, and confirm you get $T_6 = 21$. Try the same trick for $T_8$.