How_To_Be_Good_At_Math

16 NUMBERS •^ SEQUENCES AND SHAPES

Sequences and shapes

Some number sequences can be used to create shapes by using the terms

in the sequence to measure the parts of a shape, such as the lengths of its sides.

The sequence starts

with 1, shown as a

single shape.

Triangular numbers

One sequence that can be

shown as shapes is the triangular

number sequence. If we take a

whole number and add it to all

the other whole numbers that are

less than that number, we get

this sequence: 1, 3, 6, 10, 15, ...

Each of the numbers can be

shown as a triangle.

Square numbers

If we multiply each of the numbers 1, 2,

3, 4, 5 by themselves, we get this

sequence: 1, 4, 9, 16, 25, ...

We can show this number sequence

as real squares.

1 × 1 = 1 2 × 2 = 4 3 × 3 = 9 4 × 4 = 16 5 × 5 = 25

Each new number

adds a new row to

the triangle’s base

The fourth square

number is 16

We can show the

triangular sequence

by using shapes

When we add 2,

we can arrange the

shapes in a triangle.

1 + 2 = 3

Adding 3 makes a new

triangle.

1 + 2 + 3 = 6

Now we add 4 to make

a fourth triangle.

1 + 2 + 3 + 4 = 10

Adding 5 creates a fifth

triangle, and so on.

1 + 2 + 3 + 4 + 5 = 15

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