The first three patterns of a sequence are shown below. Pattern one, pattern two, and pattern three. Each pattern is made of white squares and black squares. Part a), circle the expression for the number of black squares in pattern 𝑛. The options are six 𝑛, 𝑛 plus four, three 𝑛 plus three, and two 𝑛 plus four.
There is also a part b) that we’ll come on to. So in part a), what we’re interested in is the total number of black squares in each pattern. And we can see that the number of black squares in pattern one is six. The number of black squares in pattern two is nine. And the number of black squares in pattern three is 12. Well, if in a question we are asked to find the number of something in pattern 𝑛, so in this case the number of black squares in pattern 𝑛, then what we’re looking to find is the 𝑛th term. And what we’re trying to find is the 𝑛th term of our sequence six, nine, and 12. And this is where the six, nine, and 12 refer to the number of black squares. And then, also, we’ve got 𝑛 where 𝑛 is, is the term number.
So the first term will be where 𝑛 is equal to one, second term is where 𝑛 is equal to two, and third term where 𝑛 is equal to three. So then the first thing we do is see what the difference is between each of our terms. And we can see that there is a common difference because we add three to the previous term to get the next term. So six add three is nine. Nine add three is 12. And what this tells us is that we have an arithmetic sequence. And an arithmetic sequence is a sequence, as I’ve already said, that has a common difference. And what the common difference tells us is what the coefficient of 𝑛 is going to be. So because the common difference is positive three, then we’re gonna have positive three as our coefficient of 𝑛. So we’ve got three 𝑛.
So now what we can do is beneath our table write down the values of three 𝑛, where three 𝑛 means three multiplied by 𝑛 or the term number. So we’re gonna have three multiplied by one which is three, then three multiplied by two which is six, and three multiplied by three which is nine. Well, now as we know it’s an arithmetic sequence, all we need to do is work out what the difference is from three 𝑛 to the original sequence for the first term. And if we look, we’ve got three in three 𝑛 because three multiplied by one is three. So what do we need to do to get to the six which is our first term? Well, what we need to do is add three. So therefore, we can say that our 𝑛th term is three 𝑛 add three.
And we can double check this because we can compare the other terms in three 𝑛 to the terms in sequence. So then we’ve got six add three is nine. Nine add three is 12. So this works for the other terms. And it’s worth noting at this point, when you’ve got this stage, you have to look at what you need to add or subtract. You cannot multiply or divide. So we’ve now found our 𝑛th term which is three 𝑛 plus three. And therefore, I’ve circled the correct answer which is the third answer. Okay, great, that’s part a) finished. Now, let’s move on to part b).
So for part b), will the number of black squares always be a multiple of three? Tick a box, yes or no. Give a reason for your answer.
So we’ve got our 𝑛th term which is three 𝑛 plus three. So that tells us the number of black squares in 𝑛 patterns. So let’s see if we can factorise this to help us solve this problem. Well, to factorise our expression, what we want to do is find out a common factor in three 𝑛 and three, our terms. Well, we can see that three is going to be a common factor because we have three 𝑛 and three. Now, to factorise, we want to put it into a bracket. So we’ve got three on the outside because that’s our factor. And then inside the bracket is what we need to multiply three by to make each term.
Well, to make three 𝑛, we need to multiply three by 𝑛 because three multiplied by 𝑛 is three 𝑛. And to get positive three, we’re gonna have to have positive one. And that’s because three multiplied by positive one is positive three. So now, we fully factorised. And what we’ve got is three multiplied by 𝑛 plus one. So therefore, we can say that the number of black squares will always be a multiple of three. And that’s because if three 𝑛 plus three is equal to three multiplied by 𝑛 plus one, cause we found that by factorising, then if 𝑛 is an integer, which it must be because 𝑛 is the term number, then three multiplied by 𝑛 plus one will always be a multiple of three. And that’s because three times any number is a multiple of three by definition.