Look at the pentagonal numbers. Use finite differences to determine which function represents the pattern.
2 answers:
Answer:
f(x) = 1.5x - 0.5x
Step-by-step explanation:
The function of the pattern represented by the pentagonal numbers is the sum of three triangular numbers.
The triangular number general formula
x (x + 1) / 2
For example,
The sequence
1, 3, 6, 10
*
* * *
* * * * * *
* , * *, * * *, * * * *
_____________________________
The pentagonal numbers
The sequence:
1, 5, 12, 22, 35
As shown in the picture can be divided into three triangles
Triangle 2
x (x + 1) / 2
Triangle 1 and 3 (they are triangles one unit smaller than 2)
n (n + 1) / 2
n= x-1
Replacing n
(x-1) ((x-1) + 1) / 2
(x-1) (x) / 2
(x-1) x / 2
______________
Function represents the pattern
Triangle 2 + (Triangle 1 + Triangle 3)
Triangle 1 = Triangle 3
So then,
Triangle 2 + 2* Triangle 1
x (x +1) /2 + 2* (x -1) x/2
Rearranging
0.5 x (x +1) + x(x -1)
0.5x^2 + 0.5x + x^2 -x
(0.5 x^2 + x^2) + (0.5x -x )
1.5 x^2 - 0.5 x
______
You might be interested in
Using linear function concepts, it is found that he can correctly determine the hourly rate as follows:
subtract the supply fee, 15 dollars from the total cost, then, divide the difference by the number of hours.
<h3>What is a linear function?</h3>A linear function is modeled by:
In which:
- m is the slope, which is the rate of change, that is, by how much y changes when x changes by 1.
- b is the y-intercept, which is the value of y when x = 0, which can also be interpreted as the initial value of a function that is valid only for
, such as is the case in this problem.
The flat fee is of $15, hence b = 15, then:
Philip uses the service for 3.5 hours for his spring cleaning, and his total is $232, hence when x = 3.5, y = 232, so we use it to find the hourly rate m.
Hence, the correct option is:
subtract the supply fee, 15 dollars from the total cost, then, divide the difference by the number of hours.
You can learn more about linear function concepts at brainly.com/question/24808124