A normal cake is divided into the maximum number of pieces, p, by c slices. When c = 1, 2, 3, 4, 5, and 6, the cake is divided into 2, 4, 8, 15, 26, and 42 slices respectively. What is the maximum number of pieces into which the cake can be divided when it is cut by 8 slices? Each slice extends across the entire cake.
To find the maximum number of pieces into which the cake can be divided when it is cut by 8 slices, we can look for a pattern in the given data.
Let's look at the number of slices obtained when the cake is divided by 1, 2, 3, 4, 5, and 6 slices.
When c = 1, the cake is divided into 2 slices.
When c = 2, the cake is divided into 4 slices.
When c = 3, the cake is divided into 8 slices.
When c = 4, the cake is divided into 15 slices.
When c = 5, the cake is divided into 26 slices.
When c = 6, the cake is divided into 42 slices.
From these values, we can observe that the increase in the number of slices follows a pattern. To find the pattern, we can look at the differences between consecutive values:
From dividing by 1 to dividing by 2, the increase is 4-2 = 2.
From dividing by 2 to dividing by 3, the increase is 8-4 = 4.
From dividing by 3 to dividing by 4, the increase is 15-8 = 7.
From dividing by 4 to dividing by 5, the increase is 26-15 = 11.
From dividing by 5 to dividing by 6, the increase is 42-26 = 16.
We can see that the differences between consecutive values are not consistent. Therefore, there is no linear pattern in the data.
However, we can observe another pattern. If we take the differences between consecutive differences, we get:
From 2 to 4, the increase is 4-2 = 2.
From 4 to 7, the increase is 7-4 = 3.
From 7 to 11, the increase is 11-7 = 4.
From 11 to 16, the increase is 16-11 = 5.
We can see that the differences between consecutive differences are consistent. It looks like we are increasing by 1 every time.
If we continue this pattern, the next difference would be 5 + 1 = 6.
So, if we take the difference of the last given value (42) and add the next difference (6), we can find the maximum number of pieces when the cake is divided by 8 slices:
42 + 6 = 48.
Therefore, the maximum number of pieces into which the cake can be divided when it is cut by 8 slices is 48.