Video: GCSE Mathematics Foundation Tier Pack 1 • Paper 1 • Question 11

GCSE Mathematics Foundation Tier Pack 1 • Paper 1 • Question 11

03:34

Video Transcript

A sequence of patterns is made up of shaded squares and circles. How many circles are needed to make the 10th pattern?

Let’s count the number of circles there are in each pattern. In the first pattern, we have one circle; in the second pattern, we have two circles; and in the third pattern, we have three circles. We can see that every time the pattern number increases, the number of circles in the pattern increases by one.

Now we can look at these patterns as an arithmetic sequence, where the pattern number refers to the term number or 𝑛 and the number of circles is the actual term itself. Our arithmetic sequence has a common difference of one. So let’s compare our pattern to the arithmetic sequence 𝑛.

The first three terms of the arithmetic sequence 𝑛 are one, two, and three, which is identical to the number of circles in each of our patterns. And so the arithmetic sequence which represents the number of circles in each pattern is simply 𝑛. To find the number of circles in the 10th pattern, we take our pattern number, which is 10, and set 𝑛 equal to 10. And we found that the 10th pattern requires 10 circles.

Part b) How many shaded squares are needed to make the 13th pattern?

Let’s again start by looking at how many shaded squares are in each of the three patterns. In the first pattern, there are four squares; in the second pattern, there are six squares; and in the third pattern, there are eight squares. Again, this is another arithmetic sequence. So let’s look at the common difference between the terms. And we see that we have a common difference of two.

Now let’s compare our sequence with the arithmetic sequence two 𝑛. The first term of two 𝑛 is two, the second is four, and the third is six. Comparing two 𝑛 to our sequence, you can see that we have to add two to each of the terms. So the 𝑛th term for the arithmetic sequence for the number of shaded squares in the 𝑛th pattern is two 𝑛 plus two. Therefore, to find the number of shaded squares in the 13th pattern, we simply substitute 𝑛 equals 13 into this equation. And this gives us two times 13 plus two, which is 26 plus two, which is equivalent to 28 shaded squares. So 28 shaded squares are required to make the 13th pattern.

Robyn says, “When the pattern number is even, the number of shaded squares needed will always be a multiple of three.” Part c) Is Robyn correct? You must give a reason for your answer.

Let’s look at the number of shaded squares. For the first few patterns where the pattern number is even, if you remember back to part b, we found that the number of shaded squares in the pattern 𝑛 is given by two 𝑛 plus two. So in pattern number two, we take 𝑛 equals two. This gives us that there’ll be two times two plus two, which is also equal to six shaded squares in pattern two. And six is a multiple of three. So Robyn’s statement holds for pattern number two.

Now let’s try the next even pattern number, so that’s pattern number four, where 𝑛 is equal to four. And so we’ll have two times four plus two, which is also equal to 10 shaded squares. However, 10 is not a multiple of three. And so now we have found a counter example to Robyn’s statement. Now we can conclude that Robyn is incorrect.

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