Question

Look at the pentagonal numbers. Use finite differences to determine which function represents the pattern.

f(x) = 1.5x - 0.5x

f(x) = 1.5x^2 + 1.5x

f(x) = 1.5x^2 - 0.5v

f(x) = 0.5x^2 + 0.5x

Answer 1

C on edg

Explanation:

Answer 2

f(x) = 1.5x - 0.5x

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

______