A sequence of patterns uses white tiles and grey tiles. The first five patterns are shown below. Part a) Work out the total number of tiles in pattern 50.
There is also a part b) that we will look at later. For the purpose of this explanation, the grey tiles have been shaded yellow so it is clear which ones are shaded. The first part of the question asked us to calculate the total number of tiles in pattern 50. The easiest way to start a question like this is to set up a table with the pattern number and the number of tiles. Pattern number one has a total of five tiles. Pattern number two has a total of eight tiles. Pattern number three has 11 tiles. Pattern four 14 tiles. And pattern five has 17 tiles in total.
We notice that the number of tiles increases by three with each pattern. Three extra tiles are added below the previous pattern. As the number of tiles is increasing by the same amount each time, our pattern is linear. In order to work out the number of tiles in pattern 50, we firstly need to work out the 𝑛th term formula for this pattern.
In any pattern or sequence where the common difference is the same, our 𝑛th term formula will begin with this number multiplied by 𝑛, in this case three multiplied by 𝑛 or three 𝑛. The sequence three 𝑛 generates the numbers in the three times table: three, six, nine, 12, and so on. This is because three multiplied by one is three, three multiplied by two is six, three multiplied by three is nine, and so on.
Our final step is to find a link between these numbers in the three times table and the initial sequence: five, eight, 11, 14, 17. To get from three to five, we add two. This also works to get from six to eight, nine to 11, 12 to 14, and 15 to 17. This means that our 𝑛th term formula for this pattern or sequence is three 𝑛 plus two. We can check this by substituting in any of the numbers in the table.
For example, substituting in 𝑛 equals four gives us three multiplied by four plus two. Three multiplied by four is 12. And adding two gives us 14, which was the number of tiles in pattern number four. To calculate the number of tiles in pattern 50, we need to substitute 𝑛 equals 50. This gives us three multiplied by 50 plus two. Three multiplied by 50 is 150. Adding two to this gives us 152. The total number of tiles in pattern number 50 is 152.
The second part of the question says the following. b) Complete this function machine for the sequence of patterns.
We’re trying to work out a two-step function machine that takes us from the number of grey tiles to the number of white tiles. In the first pattern, there’re no grey tiles and five white tiles. In the second pattern, we have two grey tiles and six white tiles. Pattern three has four grey tiles and seven white tiles. Pattern four has six grey tiles and eight white tiles. And pattern five has eight grey tiles and nine white tiles. The top row of each of the patterns has five white tiles. And every additional row has two grey tiles and one white tile.
This means that for every two grey tiles we add, we add one white tile. The difference between pattern one and pattern three is we have added four grey tiles and two white tiles. Two is half of four. The same is true for pattern four and pattern five. Three is a half of six and four is a half of eight. We can therefore see that the number of new white tiles is half the number of new grey tiles.
Halving a number is the same as dividing by two. Therefore, the first step in our function machine is divide by two. In every single one of our patterns, we have five white tiles in the top row. This means that the second step in our function machine is add five. We halve the number of grey tiles and then add five to calculate the number of white tiles. Let’s check each of the patterns to see if this works.
In pattern one, there was zero grey tiles. So we need to divide zero by two and add five. This is equal to five. Therefore, the function machine works for pattern one. For pattern two, there were two grey tiles. We need to divide two by two and add five. Two divided by two equals one. Adding five gives us six. Therefore, the function machine works for pattern two. For pattern three, four divided by two is two. Adding five gives us seven. So it also works for pattern three. We can continue this for patterns four and five to check that the function machine works for each of the patterns.
The number of white tiles in each pattern can be calculated by dividing the number of grey tiles by two and then adding five.