Question

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

Answer 1

Answer:

C on edg

Step-by-step explanation:

Answer 2

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  

______