Diagram # # of squares Pattern for
1 1 Cal.# unit sq.
1^2+0^2
2 5 2^+1^2
3 13 3^2+2^2
4 ? ?
10 ? ?
? 421 ?
Just need to know how to come up with the pattern?
To come up with the pattern for the number of squares in the diagram, we can analyze the given information.
Looking at the pattern, we can see that as the diagram number increases, the number of squares also increases. We can start by examining the first few values:
Diagram #1: 1 square
Diagram #2: 5 squares
Diagram #3: 13 squares
From diagram #1 to diagram #2, the number of squares increases by 4 (5 - 1 = 4).
From diagram #2 to diagram #3, the number of squares increases by 8 (13 - 5 = 8).
To find the pattern, we can observe that the difference in the number of squares from one diagram to the next is increasing by a constant amount. Based on this observation, we can infer that the pattern for the number of squares involves a quadratic equation.
Let's consider the difference in the number of squares between diagram #2 and diagram #3, which is 8. This difference corresponds to the sum of squares of two consecutive numbers (3^2 + 2^2 = 13).
From this, we can deduce a possible pattern for the number of squares:
Number of squares = diagram number^2 + (diagram number - 1)^2
Using this pattern, we can find the missing values:
Diagram #4: 4^2 + (4-1)^2 = 16 + 9 = 25 squares
Diagram #10: 10^2 + (10-1)^2 = 100 + 81 = 181 squares
Diagram ? (unknown): ?^2 + (?-1)^2 = 421 squares
To determine the value of the missing diagram number, we can equate the equation to 421 and solve for the unknown:
?^2 + (?-1)^2 = 421
?^2 + ?^2 - 2? + 1 = 421
2?^2 - 2? - 420 = 0
?^2 - ? - 210 = 0
Using the quadratic formula, we can solve for ?:
? = [(-b) ± √(b^2 - 4ac)] / (2a)
In this case, a = 1, b = -1, and c = -210. Solving the equation, we get:
? = [(-(-1)) ± √((-1)^2 - 4(1)(-210))] / (2(1))
? = (1 ± √(1 + 840)) / 2
Calculating the values:
? = (1 ± √841) / 2
? = (1 ± 29) / 2
The two possible solutions for ? are (1 + 29)/2 = 15 and (1 - 29)/2 = -14. However, since we are dealing with the number of diagrams, the value of ? cannot be negative. Therefore, the missing diagram number is 15:
Diagram #15: 15^2 + (15-1)^2 = 225 + 196 = 421 squares