A sequence of patterns made with grey and white square tiles is shown here. Draw pattern four in the space below.
To draw the next pattern, let’s start by thinking of the grey and white tiles as separate patterns in themselves. There are three grey tiles in the first pattern. Two grey tiles are added each time horizontally to the top of the pattern. The next grey pattern will, therefore, be just like the third pattern with two extra squares added to the top.
Now, let’s look at the white tiles. We started with one white tile. This time, two white tiles are added each time horizontally to the bottom of the pattern. The next white pattern will, therefore, be just like pattern three with two extra squares added to the bottom. Combining these gives the full pattern as shown. And that’s the fourth pattern completed.
Find the total number of tiles in pattern seven.
Let’s write down the number of tiles in each pattern. It can be sensible to do this in table form. In the first pattern, that’s term number one, there were three grey tiles and one white tile. That gives us a total of four tiles. In pattern two, there were five grey tiles and three white tiles. That gives us a total of eight. And in pattern three, there are seven grey tiles and five white tiles, which gives us a total of 12.
Notice the number of tiles is increasing by four each time. 12 plus four is 16, 16 plus four is 20, 20 plus four is 24, and 24 plus four is 28. This means there are a total number of 28 tiles in pattern seven.
Write, in terms of 𝑛, an expression for the number of grey tiles in pattern 𝑛.
This expression is often called the 𝑛th term. And there is a really nice way to help us remember how to find the 𝑛th term for an arithmetic sequence. If someone asks you to find the 𝑛th term, you say “DnO.” Now, obviously, this isn’t actually mean don’t know. Each letter in this mnemonic stands for something.
The D stands for difference. The first thing you do is find the difference between each term in the pattern. The n stands for 𝑛; you multiply the difference by 𝑛. And the O stands for the zero term; that’s the one that will come before the first term in the sequence.
The sequence of grey tiles begins with the numbers three, five, and seven. The D part of our mnemonic means find the difference — the common difference — between each term. The difference between three and five is two and the difference between five and seven is also two. This is called the term-to-term rule: to find the next number in the sequence, we simply add two.
So we have a common difference of two. And in fact, the end part of our mnemonic tells us to multiply two by 𝑛. So the first part of our 𝑛th term is two 𝑛. The zero term is the term that comes before the start of the sequence, before the first term.
Since this sequence is going up in twos, we can work backwards by subtracting two. Subtracting two from three and we get one. The zero term in this sequence is one. And our expression in terms of 𝑛 for the number of grey tiles in pattern 𝑛 is two 𝑛 plus one.
Now, whilst the DnO method is the most efficient way of doing it, it may not be the method that you’ve been taught. The starting point for the alternative method is still to find the common difference between the terms. We said the common difference was two. So the two 𝑛 part still stands. To find the bit that we add or subtract from the two 𝑛, we write out the two times tables.
The first number in the two times tables is two multiplied by one, which is two. The second number in the two times tables is two multiplied by two, which is four. And the third term is two multiplied by three, which is six.
We want to see what we need to do to the two times tables to get it to our sequence. If we add one to each number in the two times tables, the two 𝑛 sequence we end up at our sequence are three, five, seven. So the 𝑛th term for our sequence is two 𝑛 plus one, as we showed earlier.