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 3 of 6: Solve backward

m=1+2\left(\frac76\right)+3\left(\frac76\right)^2+\cdots+n\left(\frac76\right)^{n-1}

Detailed analysis

Rearrange the recurrence as $m_r=r+\frac76m_{r+1}$ . Starting from $m_n=n$ and iterating backward gives the displayed formula.