# Arithmetic Series Word Problem (Analysis)

#### iNET

Hi, I'm not sure if this is the right section for this question, so if not please redirect me to the correct subforum.

Q: There are k animal feeding stations arranged in a line with a supply hut. The stations are s yards apart and the nearest is t yards from the hut. An attendant carries n bags of feed, one at a time, to each feeding station. How far will he have traveled when he arrives back at the hut after servicing all stations?

Here's what I did, but I don't know if it's correct: (S.k denotes 'S of k')

S.k=(2t + 2(t+s)+...+2(t+(k-1)s))n
= n((k/2)(2(2t)+(k-1)2s))
= n((k(4t+2sk-2s))/(2))
= n((4kt+2sk^2-2sk)/(2))
= n(kt+sk^2+sk)
= nk(t+sk+s) yards

#### romsek

MHF Helper
I get

$2 n t + 2n(s+t) + 2n(2s+t) + \dots 2n((k-1) s + t) = 2 n t k+2 n s \displaystyle{\sum_{j=1}^{k-1}}j = 2 n t k + 2 n s \dfrac {k(k-1)}{2}=2 n k \left(t + \dfrac{s(k-1)}{2}\right)$

1 person

#### iNET

Thanks! I don't exactly understand the method you used to get your answer, but using your answer I was able to see the mistake in my own work that–when corrected–gave me a solution identical to yours.

#### romsek

MHF Helper
the main thing is

$\displaystyle {\sum_{j=1}^{k-1}}j =\dfrac {k(k-1)}{2}$

It looks like you set the problem up correctly. We both start from the same basic equation.

#### iNET

I see.

This may be a stupid question, but the answer could also be simplified to:

nk(2t+s(k-1))

Is one answer somehow 'more' correct than the other one, or are both answers equally acceptable?

Thanks for your help.

#### romsek

MHF Helper
I see.

This may be a stupid question, but the answer could also be simplified to:

nk(2t+s(k-1))

Is one answer somehow 'more' correct than the other one, or are both answers equally acceptable?

Thanks for your help.
they are both the same answer.

Yours is a bit simpler than mine.