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
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