10.2 Special sequences — Learn

Prerequisites: multiplication, square numbers, and term-to-term rules.

Objectives: recognise square, cube, triangular and Fibonacci sequences. This section covers A24.

Square and cube numbers

A square number is a whole number multiplied by itself. The square numbers begin $1,4,9,16,25,\ldots$, from $1^2,2^2,3^2,4^2,5^2,\ldots$.

A cube number is a whole number multiplied by itself twice. The cube numbers begin $1,8,27,64,125,\ldots$, from $1^3,2^3,3^3,4^3,5^3,\ldots$.

Example 1. Is $49$ a square number?

So $49$ is the seventh square number.

Triangular numbers

Triangular numbers count dots arranged in rows of $1,2,3,\ldots$. Add the row sizes to find each term.

Example 2. Find the first four triangular numbers.

The sequence is $1,3,6,10,\ldots$. Each new term adds the next whole number.

Fibonacci sequence

In a Fibonacci sequence, each term after the first two is the sum of the previous two terms.

Example 3. Continue $2,3,5,8,\ldots$.

The next terms are $13$ and $21$.

Watch out. A sequence can have increasing differences without being a square sequence. Check its terms against $1^2,2^2,3^2,\ldots$ rather than guessing from the first two terms.

Progress check. Pattern A has $5,8,11,14,\ldots$ tiles. It is not a square sequence: the changes are all $3$, unlike the square numbers.

You should now be able to

• recognise and generate square and cube numbers;
• build triangular numbers by adding successive whole numbers;
• continue a Fibonacci sequence by adding the previous two terms;
• explain which feature identifies each special sequence.