Method of difference part b

The terms progressively generated by the summation are

4        (4 + 9)       ( 4 + 9 + 16)      (4 + 9 + 16 + 25)     ( 4 + 9 + 16 + 25 + 36)     =

4           13               29                      54                              90          using the S of D, we have

        9             16               25                       36

               7               9                   11

                       2                 2

                                0

We have three non-zero differences so the polynomial will be a cubic in the form

ax^3 + bx^2 + cx + d

So......we have this system of equations

a(1)^3 +  b(1)^2 + c(1) + d = 4

a(2)^3 + b(2)^2  + c(2) + d  =  13

a(3)^3 + b(3)^2 + c(3) +  d =  29

a(4)^3 + b(4)^2 + c(4) +  d = 54

a  + b  +   c  +   d  =  4

8a + 4b + 2c + d = 13

27a + 9b + 3c + d = 29

64a + 16b + 4c + d = 54

Solving this system produces

a = 1/3     b = 3/2    c = 13/6   d =  0

So the polynomial is

(1/3)x^3 + (3/2)x^2 + (13/6) x

x  [ ( 2/6)x^2 + (9/6)x + (13/6) ]  

x [  2x^2 + 9x  + 13 ]

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               6

You  can sub  n for x, YEEEEEET...same difference....!!!