Feeding Chickens - Arithmetical Progression

Date: 7/6/96 at 20:53:51
From: Wong Cheong Siong
Subject: Feeding Chickens

Dear Dr. Math,

1) On a certain date a farmer has 3000 hens. One week later he 
sells 20 and he continues to do this each week.If each hen costs 8p 
per week to feed, calculate the total cost of feeding the hens up to 
the day when his stock reaches 500.

2) A farmer, starting with 2100 hens, sells ten at the end of the 
first week... after how many weeks will he have sold all his hens?

Best Regards,
Wong Cheong Siong

Date: 7/8/96 at 11:24:12
From: Doctor Brian
Subject: Re: Feeding Chickens

1) For week one, he has 3000 hens, so it costs 3000 * 8p
For week two, he has 2980 hens, so it costs 2980 * 8p

We need to sum up:
8p * (3000 + 2980 + 2960 + ... + 500)

The sum in parentheses is arithmetic, first term = 3000, 
difference = -20. It's not too tough to show that there are 126 terms.
One formula for the sum is (half the # of terms) * (sum of end terms)
which would give us 63 * 3500 = 220500, which when multiplied by 8p,
gives a total cost of 176400p.

*I don't know how to convert foreign money, so you're on your own 
there*

-Doctor Brian,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 7/8/96 at 5:53:55
From: Doctor Anthony
Subject: Re: Feeding Chickens

1) The cost of feeding 3000 hens for a week is .08*3000 = 240 UKP
(UKP = pounds sterling)

The number of weeks is (3000-500)/20 = 125
Cost of feeding 500 hens for a week = .08*500 = 40 UKP
Total cost = n/2(first + last) = (125/2)(240 + 40)
                               = (125/2)*280
                               = 17500 UKP

2) This is an AP with a=10, d= 10  Sn = 2100,  n is to be found

Solve the equation (n/2)(2a + (n-1)d) = 2100

               (n/2)(20 + (n-1)*10) = 2100

                         n(2 + n-1) = 420
                       2n + n^2 - n = 420

                      n^2 + n - 420 = 0
                       (n+21)(n-20) = 0  we can't have negative n, 

           So we must have n = 20 weeks.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/