MathDB

Consider a table of cyclic permutations (

n\ge2

)

\begin{matrix} 1, & 2, & \ldots, & n-1, & n \\ 2, & 3, & \ldots, & n, & 1, \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ n, & 1, & \ldots, & n-2, & n-1. \end{matrix}

Then multiply each number of the first row by that number of the

k

-th row that is in the same column. Sum all these products and denote

s_k

the result (e.g.

s_2=1\cdot2+2\cdot3+\cdots+(n-1)\cdot n+n\cdot1

). a) Find a recursive relation for

s_k

in terms of

s_{k-1}

and determine the explicit formula for

s_k

. b) Determine both an index

k

and the value of

s_k

such that the sum

s_k

is minimal.

Problem Statement