a To find the number of cubes when n=4, we must identify a pattern. Notice that the bigger cube's side equals the corresponding value of n. Therefore, n=4 correspond to a big cube with a side of 4 units.
However, notice that in each figure, there is always one cube missing from the top corner in the front. This should be the case for n=4 as well.
To obtain the number of small cubes when n=4, we must calculate the big cube's volume in terms of the smaller cubes and then subtract 1 to account for the smaller missing cube. Our formula becomes the following.
n^3-1
Let's apply this formula when n=4.
Therefore, n= 1 must correspond to a big cube with a side of 1 unit. But since the volume of the figure is 1 less than this, we end up with no cube at all.
1^3-1=0
c From Part A, we found the general equation for the pattern. a_n=n^3-1
We also know that when n=1 there is no cube at all. Let's confirm this and also calculate the number of cubes when n=5.
|c|c|c|
n & n^3-1 & =
1 & 1^3-1 & 0
5 & 5^3-1 & 124
There are 124 cubes in figure n=5.
Arithmetic: &Constant difference
&between consecutive terms.
&t(n)=d(x-1)+t(0) [0.7em]
Geometric: &Constant factor
&between consecutive terms.
&t(n)=t(1)r^(n-1)
Examining the equation from Part A, we notice that the sequence cannot be arithmetic as it contains a cubed variable. Also since the variable is in the base of a power, it cannot be geometric either. Therefore, its neither.
