2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 7

Problem

Let $k$ be a natural number. Show that the sum of the $k^{th}$ powers of the first $n$ positive integers is a polynomial of degree $k + 1$ , i.e., $1^k + 2^k + 3^k + \cdots + n^k = p_{k+1}(n)$ , where $p_{k+1}(t)$ is a polynomial of degree $k + 1$ . For example, for $k = 1$ we have

$1 + 2 +\cdots + n = \sum_{j=1}^{n} {j} = \frac{n(n+1)}{2}=\tfrac{1}{2}n^2+\tfrac{1}{2}n$

hence $p_2(t) = \tfrac{1}{2} t^2 + \tfrac{1}{2}t.$

Solution

By Faulhaber's Formula, $1^k+2^k+\cdots+n^k=\frac{1}{k+1}\sum_{i=0}^k {k+1\choose i}b_i n^{k+1-i}$ where the $b_i$ are given constants beyond the scope of this proof (the Bernoulli rationals). Clearly, considering the case $i=0$ gives a term containing $n^{k+1}$ (note that $b_0=1\ne0$ ), and all smaller terms contain powers of $n$ with exponent less than $k+1$ ; therefore, $p_{k+1}(t)$ is indeed a polynomial of degree $k+1$ , and we are done.

~ eevee9406

2014 UNM-PNM Contest II (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
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2014 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 7

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