The diagram shows a pattern made from triangles. Part a, work out the number of triangles in pattern seven. Part b, complete the rule below. The number of triangles equals the pattern number times blank minus blank. Part c, calculate the pattern number that has 121 triangles.
We notice that in pattern one, we have one triangle. In pattern two, we have three triangles. In pattern three, we have five triangles. And in pattern four, we have seven triangles. Each time, we’re increasing by two. Part a wants to calculate pattern seven, the number of triangles in pattern seven. We can calculate this by adding two to pattern five then to pattern six then to pattern seven. Since pattern four had seven triangles, pattern five will be made of nine triangles. Pattern six would be made of 11 triangles. And pattern seven would be made of 13 triangles. This means the answer to part a is 13 triangles.
In part b, we need to complete the rule, the number of triangles equals the pattern number times something minus something. Let’s plug in what we know about pattern two. There are three triangles, and the pattern number is two. If we look at pattern three, it has five triangles and three is its pattern number. If we plug in three for the multiplication box, we would have two times three which is six. And we would have to subtract three to get three. Now, two times three minus three is equal to three. But we have to use the same values for every pattern. So now we need to know, is three times three minus three equal to five? Three times three equals nine. Nine minus three equals six. It does not make the second statement true.
For these type of sequences, we can look at what we’re adding each time. This will show us what we should multiply by each pattern number. Since we’re adding two every time, we need to multiply our pattern number by two. Two times two is four, minus one is three. Two times three is six, minus one is five. This means that our rule is, the number of triangles are equal to the pattern number multiplied by two minus one.
In part c, we want to calculate the pattern number if we know the number of triangles. We know the number of triangles 121. We can use the variable 𝑛 for the pattern number. We know that we multiply two by 𝑛 and then subtract one. Now we can solve this equation. 121 equals two 𝑛 minus one. And so, we add one to both sides, which gives us 122 equals two 𝑛. From there, we divide both sides by two. 122 divided by two is 61. Two 𝑛 divided by two equals 𝑛. Since 𝑛 represents the pattern number, our pattern number should be 61. At pattern number 61, there would be 121 triangles.