Step-by-step explanation:
Janyce's rule doesn't work. In the case where n = 3 her rule predicts that there will be 3(3) - 1 = 8 squares but instead, there are 10 squares. Likewise, for n = 4, there are 17 squares instead of 11 that she predicted. Instead, I propose the rule [tex]n^2 + 1.[/tex] It works for all the figures shown and mostly likely, for all other values of n.
2. Part A;
Given that the number of squares obtained using Janyce conjecture for
figures 3 and 4 which are 8 and 11 squares respectively and the number of
squares in the diagram for figures 3 and 4 which are 10 and 17 squares
respectively are not the same, disproves Janyce conjecture
Part B:
The rule for the number of squares in the nth squares is n² + 1
The reasons the above conclusion and rule are correct is as follows:
2. Part A
The given number of squares in each figure are;
[tex]\begin{array}{|c|c|c|}\mathbf{Figure \ number}&\mathbf{Number \ of \ squares} &\mathbf{Janyce \ conjecture \ (3\cdot n - 1)}\\&&\\1&2 \ squares &(3 \times 1 - 1 = 2) \ squares \\2&5 \ squares&(3 \times 2 - 1 = 5) \ squares\\3&10 \ squares &(3 \times 3 - 1 = 8) \ squares\\4&17 \ squares&(3 \times 4 - 1 = 11) \ squares\end{array}[/tex]
Given that the number squares in figure 3, and 4 are;
Figure 3 = 10 squares
Figure 4 = 17 squares
Janyce conjectured number of squares are;
Figure 3 = 8 squares
Figure 4 = 11 squares
The values for the number of squares in the diagram in figures 3 and 4 are different from Janyce conjecture which disproves Janyce's conjecture for the number of squares in the nth figure
Part B
From the diagrams, we have;
The number of squares on the each figure is given by a square shape with a side length equal to the figure number of squares plus one extra square at the top right end corner, we have;
The area of the square part of the shape = n² = The number of squares in the square shape
The number of squares in the nth figure = n² + 1
Verifying, we have;
Figure 1 = (1² + 1) squares = 2 squares (Same as in diagram)
Figure 2 = (2² + 1) squares = 5 squares (Same as in diagram)
Figure 3 = (3² + 1) squares = 10 squares (Same as in diagram)
Figure 4 = (4² + 1) squares = 17 squares (Same as in diagram)
Therefore;
The correct formula for the number of squares in the nth figure = n² + 1
Learn more about the nth term of a sequence here:
https://brainly.com/question/12998719
https://brainly.com/question/15325765
https://brainly.com/question/21493252
