Question

if f(x)=ax^3+bx^2+cx+d find f(1), f(2), f(3), f(4), f(5), f(6) and show that the third differences of these values are constant.

Answer 1

third differences are all 6a

Explanation:

Substituting for x, we find the sequence values to be ...

(a+b+c+d), (8a+4b+2c+d), (27a+9b+3c+d), (64a+16b+4c+d), (125a+25b+5c+d),

(216a+36b+6c+d)

Then first differences are ...

(7a+3b+c), (19a+5b+c), (37a+7b+c), (61a+9b+c), (91a+11b+c)

Second differences are ...

(12a+2b), (18a+2b), (24a+2b), (30a+2b)

And the third differences are ...

(6a), (6a), (6a) . . . . . constant

_____

Additional comment

These results can help you write a polynomial relation up to 3rd degree for a sequence that has constant differences at any level up to 3rd differences.

Of course, we find the differences by subtracting each term from the one following.