Problem 6

In a sports contest,

m

medals were awarded on

n

successive days (

n>1

). On the first day, one medal and

\frac17

of the remaining

m-1

medals were awarded. On the second day, two medals and

\frac17

of the then remaining medals were awarded, and so on. On the

n

-th and last day, the remaining

n

medals were awarded. Find

m

and

n

Step 2 of 6: Write the recurrence

m_{r+1}=m_r-\left(r+\frac{m_r-r}{7}\right)=\frac67(m_r-r)

Detailed analysis

For $r<n$ , day $r$ awards $r+(m_r-r)/7$ prizes. Subtracting this from $m_r$ yields $m_{r+1}=\frac67(m_r-r)$ .