Successive Differences of Powers

$a_{n}$	$a_{n - 1}$	$a_{n} - a_{n - 1}$	Difference
$288$	$121.5$	$288 - 121.5$	$166.5$
$562.5$	$288$	$562.5 - 288$	$274.5$
$972$	$562.5$	$972 - 562.5$	$409.5$
$1543.5$	$972$	$1543.5 - 972$	$571.5$

a_{n}

a_{n - 1}

a_{n} - a_{n - 1}

Difference

288

121.5

288 - 121.5

166.5

562.5

288

562.5 - 288

274.5

972

562.5

972 - 562.5

409.5

1543.5

972

1543.5 - 972

571.5

Consider the difference between the

n^{th}

and the

(n - 1)^{th}

terms of the sequence

a_{n} .

a_{n} - a_{n - 1} = a_{1} n^{p} - a_{1} (n - 1)^{p} ⇓ a_{n} - a_{n - 1} = a_{1} [n^{p} - (n - 1)^{p}]

Notice that the expression in brackets is the difference of

p^{th}

powers, which can be factored as follows.

x^{p} - y^{p} = (x - y) (x^{p - 1} + x^{p - 2} y + \dots + x y^{p - 2} + y^{p - 1})

Using this rule, the right hand-side of the equation can be simplified.

a_{n} - a_{n - 1} = a_{1} [n^{p} - (n - 1)^{p}]

$x^{p} - y^{p} = (x - y) (x^{p - 1} + x^{p - 2} y + \dots + x y^{p - 2} + y^{p - 1})$

a_{n} - a_{n - 1} = a_{1} [(n - (n - 1)) (n^{p - 1} + n^{p - 2} (n - 1) + \dots + n (n - 1)^{p - 2} + (n - 1)^{p - 1})]

Distribute $- 1$

a_{n} - a_{n - 1} = a_{1} [(n - n + 1) (n^{p - 1} + n^{p - 2} (n - 1) + \dots + n (n - 1)^{p - 2} + (n - 1)^{p - 1})]

Subtract term

a_{n} - a_{n - 1} = a_{1} [(1) (n^{p - 1} + n^{p - 2} (n - 1) + \dots + n (n - 1)^{p - 2} + (n - 1)^{p - 1})]

$a \cdot 1 = a$

a_{n} - a_{n - 1} = a_{1} [n^{p - 1} + n^{p - 2} (n - 1) + \dots + n (n - 1)^{p - 2} + (n - 1)^{p - 1}]

This final equation shows that taking differences reduces the degree of the polynomial by

1 .

If taking the differences of consecutive terms continues, after the

p^{th}

differences the degree of the polynomial will be

0 .

Therefore, the polynomial will be a constant.

Conclusion If a polynomial has degree p, then the p^{th} differences are a nonzero constant .