The first three patterns in a sequence are shown. Part a) How many black squares are in pattern six? Louise says, “There are four white squares in pattern three, so there are eight white squares in pattern six.” Part b) Is Louise right? Give a reason for your answer.
In order to answer these questions, we could draw pattern four, pattern five, and pattern six. However, this might take a long time in the exam. An alternative method would be to draw a table and try and find the pattern.
The first pattern has three black squares, the second pattern has six black squares, and the third pattern has nine black squares. As these numbers have a common difference of three, we can see that the number of black squares is increasing by three with each new pattern. Adding three to nine gives us 12. 12 plus three equals 15, and 15 plus three equals 18. We can therefore say that there are 18 black squares in pattern six.
You might also notice that the number of black squares is the three times table: three, six, nine, 12, and so on. Therefore, the 𝑛th term formula for the number of black squares is three 𝑛 or three multiplied by 𝑛. To work out the number of black squares in the sixth pattern, we would multiply three by six. Three multiplied by six is equal to 18, which is the answer that we got previously.
The second part of our question related to the white squares. Louise said that there are four white squares in pattern three, so there are eight white squares in pattern six. We need to prove whether Louise is right or wrong.
We can answer this by setting up the table in the same way as in part a. There are two white squares in pattern one, there are three white squares in pattern two, and there are four white squares in pattern three. The common difference this time is one. Therefore, for each new pattern, we have one extra white square.
Pattern four will have five white squares. There will be six white squares in pattern five. And there will be seven white squares in pattern six. This means that Louise is not right. There will be seven white squares in pattern six, not eight squares as Louise predicted.
We can also see from the table that the number of white squares is one more than the pattern number. Therefore, the 𝑛th term formula for the white squares would be 𝑛 plus one. In the sixth pattern, we add one to six to give us the number of white squares. Six plus one equals seven. Using these 𝑛th term formulas can help us work out the number of black or white squares in any pattern in the sequence.