Each diagram in the sequence is obtained by drawing of a 1 unit square on each side that forms the perimeter of the previous diagram, the 1st digram contains (1) square, the 2nd diagram (4) squares and the 3rd diagram (13) squares, I would like to know how to find how many squares will be in the fourth diagram? and what are the appropriate values for the rows marked 1st, 2nd and 3rd?
shouldnt the 2nd diabram have 5 squares.
I believe the CHANGE in the number of squares grows by 4 each time. So, 1, 5, 13, 25, and so on.
or
the total number of squares after n stages is
2n^2 - 2n + 1
test: if n= 4
we get 2(16) - 8 + 1 = 25
as economyst predicted
To find the number of squares in the fourth diagram, we need to understand the pattern in the sequence.
Let's analyze the given information about the sequence:
- The first diagram contains 1 square.
- The second diagram contains 4 squares.
- The third diagram contains 13 squares.
From these initial values, we can observe that the number of squares is increasing at an increasing rate.
To determine the number of squares in the fourth diagram, we need to establish a pattern.
If we look closely at the sequence, we can notice that the increase from one diagram to the next seems to be following a quadratic pattern. Specifically, it seems to be increasing by a number equal to the square of the diagram number.
Let's calculate the pattern:
- In the first diagram, we have 1 square.
- In the second diagram, we have 1 + 2^2 = 1 + 4 = 5 squares.
- In the third diagram, we have 5 + 3^2 = 5 + 9 = 14 squares.
Following this pattern, we can calculate the number of squares in the fourth diagram:
- In the fourth diagram, we should have 14 + 4^2 = 14 + 16 = 30 squares.
Therefore, the fourth diagram will contain 30 squares.
Regarding the appropriate values for the rows marked 1st, 2nd, and 3rd:
- In the first diagram, the number of squares is 1.
- In the second diagram, the number of squares is 4.
- In the third diagram, the number of squares is 13.
- In the fourth diagram, the number of squares is 30.
Hence, the appropriate values for the rows marked 1st, 2nd, and 3rd are 1, 4, and 13, respectively.