Question 1190223
<pre>
What a gruesome problem! (Men getting killed in war!)

Let K = the number of units of provisions (k-rations] required for 5700 
men for 66 days.

Then K/2700 = the number of units of provisions required for 1 man for 
66 days.
 
Then K/(2700•66) = the number of units of provisions required for 1 man 
for 1 day.

If nobody were killed, the sequence over 66 days would be this
obvious sum of 66 terms with all identical terms:

{{{
5700*(K/(5700*66))+
5700*(K/(5700*66))+
5700*(K/(5700*66))+
""*""*""*""+5700*(K/(5700*66))}}}{{{""=""}}}{{{K}}}

However, 20 men are killed each day, so the sequence has more terms and
goes like this:

{{{
5700*(K/(5700*66))+
5680*(K/(5700*66))+
2660*(K/(5700*66))+
""*""*""*""+5700*(K/(5700*66))}}}{{{""=""}}}{{{K}}}

We want to find out how many terms the sequence must have to make 
the above equation true. 

We can divide both sides by K

{{{
5700*(1/(5700*66))+
5680*(1/(5700*66))+
5660*(1/(5700*66))+
""*""*""*""+5700*(1/(5700*66))}}}{{{""=""}}}{{{1}}}

We can factor out {{{1/(5700*66)}}}

{{{
(1/(5700*66))*(5700+5680+5660+""*""*""*""+"?")}}}{{{""=""}}}{{{1}}}

Suppose the sequence has N terms, then its sum is 

       {{{S[N]=(N/2)(2a[1]+(N-1)d))}}}, where a<sub>1</sub>=2700, and d=-20

{{{(1/(5700*66))*(N/2)(2*5700+(N-1)(-20))}}}{{{""=""}}}{{{1}}}

Factor 20 out of the last parentheses on the left

{{{(1/(5700*66))*(N/2)(20)(2*285-(N-1))}}}{{{""=""}}}{{{1}}}

Further simplifying:

{{{(N/37620)(570-N+1)}}}{{{""=""}}}{{{1}}}

{{{N(571-N)}}}{{{""=""}}}{{{37620}}}

{{{571N-N^2}}}{{{""=""}}}{{{37620}}}

{{{-N^2+571N-37620}}}{{{""=""}}}{{{0}}}

{{{N^2-571N+37620}}}{{{""=""}}}{{{0}}}

{{{(N-76)(N-495)}}}{{{""=""}}}{{{0}}}

N-76 = 0;     N-495 = 0
   N = 76;        N = 495

495 is extraneous.

Answer 76 days.

Edwin</pre>