In a sports contest there were m medals awarded on n successive days (n > 1).

On the first day 1 medal and 1/7 of the remaining (m - 1) medals were awarded.

On the second day 2 medals and 1/7 of the now remaining medals was awarded; and so on.

On the nth and last day, the remaining n medals were awarded.

How many days did the contest last, and how many medals were awarded altogether?

On day n there were n medals before the ceremony.

On day n-1 there were (7/6)*n+(n-1) medals.

On day n-2 there were (7/6)*((7/6)*n + (n-1)) + (n-2) medals.

On day n-3 there were (7/6)*((7/6)*((7/6)*n + (n-1)) + (n-2)) + (n-3) medals.

.

.

.

On day 2 there were (7/6)*...*((7/6)*((7/6)*n + (n-1)) + (n-2)) + (n-3)) + ... + 2 medals.

On day 1 there were (7/6)*((7/6)*...*((7/6)*((7/6)*n + (n-1)) + (n-2)) + (n-3)) + ... + 2) + 1 = m medals.

Multiplying out yeilds m = (7/6)^(n-1)*n + (7/6)^(n-2)*(n-1) + ... + (7/6)^2*3 + (7/6)*2 + 1.

m can also be rewritten as:

m=((7/6)^(n-1) + (7/6)^(n-2) + ... + (7/6)^3 + (7/6)^2 + (7/6) + 1)

+ ((7/6)^(n-1) + (7/6)^(n-2) + ... + (7/6)^3 + (7/6)^2 + (7/6))

+ ((7/6)^(n-1) + (7/6)^(n-2) + ... + (7/6)^3 + (7/6)^2)

.

.

.

+ ((7/6)^(n-1) + (7/6)^(n-2))

+ ((7/6)^(n-1))

Each of the summations in this representation is a geometric series.

m=( (7/6)^(n-1)*(1-(6/7)^(n) )/(1-(6/7))

+ ( (7/6)^(n-1)*(1-(6/7)^(n-1) )/(1-(6/7))

+

.

.

.

+ ( (7/6)^(n-1)*(1-(6/7)^2 )/(1-(6/7))

+ ( (7/6)^(n-1)*(1-(6/7) )/(1-(6/7))

Factoring out the common term 7*(7/6)^(n-1)

m = 7*(7/6)^(n-1) * ( 1-(6/7)^(n) + 1-(6/7)^(n-1) + ... + 1-(6/7)^2 + 1-(6/7) )

There are two series in the right factor, a geometric and a constant series.

m = 7*(7/6)^(n-1) * (n - (6/7)^n*(1-(7/6)^n)/(1-(7/6)) )

Simplifying yeilds

m = 7*(7/6)^(n-1) * (n - (-6)*(6/7)^n - -(-6)*(6/7)^n*(7/6)^n)

m = 7*(n-6)*(7/6)^(n-1) + 36

(n-6)*7^n

m = --------- + 36

6^(n-1)

n=1 and n=6 are the only integers for which the value of m is an integer.

Since the ceremony lasted more than one day, the ceremony lasted 6 days and 36 medals were given out.

*Edited on ***January 29, 2004, 2:47 pm**