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Medals (Posted on 2004-01-29) Difficulty: 3 of 5
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?

See The Solution Submitted by Ravi Raja    
Rating: 3.2000 (5 votes)

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Solution | Comment 2 of 5 |
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
  Posted by Brian Smith on 2004-01-29 14:46:46
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