Pearl made this sequence of patterns out of sticks. We can see that the first pattern has just a square of sticks, four of them. The second pattern has the sticks in a cross. And the third pattern has them in the shape of a larger cross or plus sign. Part a) tells us to find an expression, in terms of 𝑛, for the number of sticks that will be in the 𝑛th pattern in the sequence.
Bradley has 90 sticks. He is going to make a pattern in Pearl’s sequence using as many of his sticks as possible. Part b) asks us, “how many sticks will Bradley have left?”
Let’s focus on part a) to begin with. We need to find the number of sticks in the 𝑛th pattern. And so first, we need to understand how the pattern is growing in the sequence. One thing we can do to help us is to create a table of 𝑛 versus the number of sticks in the 𝑛th pattern.
So for example, in this cell of the table here, we want the number of sticks in the first pattern. And we can see that there are four sticks in the first pattern arranged in a square. So here, we put four. Similarly, we count the number of sticks in the second pattern. One way to do this is to see the second pattern somehow growing from the first pattern.
We can get the second pattern from the first pattern by adding three more sticks at the top, three more sticks to the right, three more sticks below, and three sticks to the left. Therefore, the second pattern has three times four which is 12 more sticks than the first pattern. And of course, four plus 12 is 16.
We can do something similar to get the third pattern from the second pattern. We can imagine that each lim of the plus sign being extended, which again that means adding 12 sticks. And 16 plus 12 is 28. Of course, we could have counted each stick one by one, but we would have missed out on how the pattern is growing.
Continuing the sequence then, we can see that the fourth pattern should have its limits extended even further. And of course, this means adding 12 more sticks, bringing us up to 40 for the fourth pattern in the sequence. Now, we see that we can just use the table to find the number of sticks in the fifth pattern and then the sixth pattern, and so on.
Each pattern has 12 more sticks than the one before it. And so we recognize this sequence as an arithmetic progression. Part a) of the question asks us to find the number of sticks in the 𝑛th pattern, which is the 𝑛th term of this arithmetic progression. The way to find the 𝑛th term of the progression is to first find the zeroth term of the progression.
What is this term? Well, we can use our common difference of plus 12 to tell us that whatever this zeroth term is, 12 more than it will be the first term, four. If we call the zeroth term 𝑥, then 𝑥 plus 12 is four. Subtracting 12 from both sides, we see that 𝑥 is four minus 12. Maybe you could have seen that without using algebra. And four minus 12 is negative eight. That’s the zeroth term.
Okay, the first term four is this zeroth term negative eight plus the common difference 12. The second term 16 is this zeroth term of negative eight plus two times the common difference 12. The third term is negative eight plus three times 12. And the fourth term is negative eight plus four times 12. And so the 𝑛th term is negative eight plus 𝑛 times 12, which we can also write as 12𝑛 minus eight. This is the number of sticks in the 𝑛th pattern in the sequence.
Now, let’s look at part b. We’re still dealing with the same sequence of patterns. Bradley has 90 sticks to play with. And he wants to use as many of those sticks as possible to make a pattern in Pearl’s sequence. In other words, he wants to make the biggest pattern possible. The question is how many of his 90 sticks will not be used in the creation of this pattern.
Okay, let’s first find which pattern he can make. One way to do this will be to continue the table we made in part a) of the question to find the number of sticks in the fifth pattern, the sixth pattern, and so on and hence whether Bradley could make that pattern, whether he has enough sticks to do so.
That method is perfectly fine and will get you the right answer. But we’re going to use a different method, involving the expression for the number of sticks in the 𝑛th pattern that we found in part a.
He has 90 sticks and so he can only build a pattern if the number of sticks in that pattern is less than or equal to 90. And so we can set up an inequality, which tells us for which 𝑛 he has enough sticks to make the 𝑛th pattern. We solve this inequality in the normal way. We add eight to both sides getting a 12𝑛 minus eight plus eight, which is just 12𝑛 on the left-hand side and 90 plus eight which is 98 on the right-hand side.
Dividing both sides by 12, we get that 𝑛 is less than or equal to 98 over 12. And putting this into our calculator, we find that 𝑛 must be less than or equal to eight and one-sixth for him to have enough sticks to build the 𝑛th pattern in the sequence.
We are interested only in whole number values of 𝑛. This inequality tells us that he can build the eighth pattern as eight is less than or equal to eight and one-sixth. But he can’t build the ninth pattern as nine is not less than or equal to eight and one-sixth. This is the same answer we would have got by extending our table.
He makes the eighth pattern then as this is the pattern he can make, which uses as many of his sticks as possible. Now, how many sticks does he use to make this pattern? Luckily, we know how many sticks there are in the 𝑛th pattern and so we can just substitute eight for 𝑛.
Doing this, we find that there are 12 times eight minus eight, which according to our calculator is 88 sticks used in the creation of the eighth pattern. And this agrees with the value we get by continuing our table.
But we are interested in how many sticks Bradley will have left. That’s how many sticks aren’t used. He has 90 sticks. And so removing the 88 sticks which we used, there are two sticks left. 90 minus 88 is two.