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

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See also

2014 UNM-PNM Contest II (ProblemsAnswer KeyResources)
Preceded by
Problem 6
Followed by
Problem 8
1 2 3 4 5 6 7 8 9 10
All UNM-PNM Problems and Solutions
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