I need a bit of help on a geomtric sequence problem. A barn contains 50 tonnes of hay. During the first week, the farmer removes 840kg of hay from the bard. In each subsequent week, she removes from the barn 96kg hay more than she removed the previous week. Work out how much hay is left in the barn after 24 weeks.
Now i know I need to come up with a geometric sequence to solve this but I'm struggling with finding the ratio. normally i'd do this by n-(n-1). but in this case each term doesn't decrease by a constant ratio.
in week 1 she removes 840kg to leave 41960kg of hay in the barn. week 2 she takes away 840kg+96kg and week 3 its 840kg+96kg+96kg and so forth. How do i go about formulating a sequence as I am used to finding sequence where each term increases/decreases by a constant. Any help would be greatly appreciated
Best Answer
I think what you want is an arithmetic sequence.
Hopefully this can get you started:
In this case, during week $k$ she will remove $H(k) = H(k-1) + 96$ and we have that $H(1) = 840$
We are interested in $50,000 - \sum_{k=1}^{24} H(k)$ bales of hay left after the 24 weeks.
Notice the following: $$H(3) = H(2) + 96 = (H(1) + 96) + 96 = H(1) + 2*96 = 840 + 2\times96$$
In general, $H(k) = 96(k-1) + 840$. This can be seen because
$$H(k+1) = 96(k) + 840 = 96(k-1+1) + 840 = 96(k-1) + 840 + 96= H(k) +96$$
We are then interested in $$50,000 - \sum_{k = 1}^{24}[96(k-1) + 840] = 50,000 - (24)(840) - 96\sum_{l=1}^{23}l$$. Where $l = k-1$