Number sequences

A sequence is just a list of numbers written in order. Some sequences are so important and so beautiful that mathematicians have given them names. Let us meet a few.

Concept

The simplest sequence is the list of counting numbers:

$1, 2, 3, 4, 5, 6, \dots$

The dots at the end mean "and it keeps going forever". Each new term is found by adding $1$ to the one before. Easy , but already powerful, because every other sequence we build comes out of this one.

Pick every second term starting from $2$ and you get the even numbers:

$2, 4, 6, 8, 10, 12, \dots$

Pick the ones in between and you get the odd numbers:

$1, 3, 5, 7, 9, 11, \dots$

Both grow by $2$ each step, but they start at different places.

Now imagine you arrange counting numbers in a triangle of dots , one dot in the first row, two in the second, three in the third, and so on. The total number of dots in the first $n$ rows gives the triangular numbers:

$1, 3, 6, 10, 15, 21, 28, \dots$

These come from adding: $1$, then $1+2$, then $1+2+3$, and so on. Triangular numbers were known to the ancient Greeks, who actually used pebbles on a beach to study them.

If instead you arrange dots in a square , $1 \times 1$, then $2 \times 2$, then $3 \times 3$ , you get the square numbers:

$1, 4, 9, 16, 25, 36, \dots$

These are $1^2, 2^2, 3^2, 4^2, \dots$ and they grow much faster than triangular numbers.

Then there are the powers of 2, which double at every step:

$1, 2, 4, 8, 16, 32, 64, \dots$

A grain of rice doubled on each square of a chessboard would, by square $64$, give more rice than the world has ever produced. Doubling sequences are sneaky , they look slow at first but become huge in a hurry.

A truly magical sequence is the Fibonacci sequence:

$1, 1, 2, 3, 5, 8, 13, 21, 34, \dots$

Each term is the sum of the two before it. Fibonacci numbers turn up in the spirals of pinecones, sunflower heads, and pineapples , a real-world pattern hidden inside biology.

There are also the primes:

$2, 3, 5, 7, 11, 13, 17, 19, 23, \dots$

These are numbers with exactly two divisors , $1$ and themselves. You will study them properly in the Prime Time chapter, but it is worth knowing that they are the "building blocks" of all whole numbers.

Worked examples

Example 1. Write the next three terms of the triangular numbers $1, 3, 6, 10, 15, \dots$.

• Each new term adds one more than the last gap added: gaps are $2, 3, 4, 5, \dots$
• $15 + 6 = 21$, $21 + 7 = 28$, $28 + 8 = 36$.
• Next three: $21, 28, 36$.

Example 2. Is $49$ a square number? A triangular number?

• $49 = 7 \times 7 = 7^2$, so yes, square.
• Triangulars near $49$: $\dots, 36, 45, 55, \dots$. So $49$ is not triangular.

Example 3. Find the $6$th term of the Fibonacci sequence $1, 1, 2, 3, 5, 8, \dots$.

• The list shows: $1$st $= 1$, $2$nd $= 1$, $3$rd $= 2$, $4$th $= 3$, $5$th $= 5$, $6$th $= 8$.
• So the $6$th term is $8$.

Example 4. A sequence starts $5, 8, 11, 14, \dots$. Find the $10$th term.

• Each step adds $3$.
• From the $1$st term to the $10$th, we add $3$ nine times: $5 + 9 \times 3 = 5 + 27 = 32$.
• So the $10$th term is $32$.

Try it yourself

1. Write the first ten odd numbers.
2. Find the $7$th triangular number.
3. Find the $8$th square number.
4. What is $2^{10}$ , the $11$th term in the powers-of-$2$ sequence starting from $1$?
5. Continue Fibonacci to ten terms.
6. Is $100$ a square? Is $100$ a triangular number?
7. Make up your own sequence with a clear rule, and challenge a friend to find the rule.

Activity

Pebble triangles. Collect $36$ small stones, beads, or buttons. Arrange them into the biggest triangle you can , $1$ in the top row, $2$ in the next, and so on. How many rows do you get? What triangular number is that? Now arrange them as the biggest square you can , what do you find? Compare the two arrangements and explain to a friend why $36$ shows up in both.