Answer:
Part A
aₙ = 5 + (n - 1)·4
Part B
y = 4·x + 1
Step-by-step explanation:
Part A
The number of orange blocks in Figure 1 = 5
The number of orange blocks in Figure 2 = 9
The number of orange blocks in Figure 3 = 13
Therefore, we get;
5, 9, 13, ... with increments of 4
The first term, a = 5
The common difference, d = 4
The sequence is in the form of an Arithmetic Progression, therefore, we have;
aₙ = 5 + (n - 1)·4
Where;
n = The Figure number
When n = 3, we have;
a₃ = 5 + (3 - 1) × 4 = 12 orange blocks as in the diagram in the question
Part B
To express the number of blocks using the linear function, y = m·x + b, we have;
Let 'x', represent the number of each figure, let 'y' represent the total number of blocks in each figure, let 'd' represent the initial amount of blocks present when x = 0, and let 'm' represent the rate of change of 'y' with each change in 'x'
We note that the rate of change the total number of blocks for each increase in figure number is 4, therefore, we have;
The rate of change, m = 4
∴ y = 4·x + d
When x = 1, y = 5, we get;
5 = 4 × 1 + d
∴ d = 5 - 4 = 1
d = 1
The linear equation that gives the number of blocks is therefore;
y = 4·x + 1.