# How do I solve this mushroom-related arithmetic sequence problem?

I interpreted this as the side of the following arithmetic sequence starting at length 1 and increasing by $$2$$ every day.

$$100 = 1 +(n-1)\cdot2$$

$$n = 99/2 +1$$

$$n = 50.5$$

I then interpreted this as the field being full by the $$51$$st day from Monday. The question asks for the number of days until the field is covered including Wednesday. Therefore, I subtracted $$2$$ from $$51$$ to get $$49$$ days as my answer. Yet this obviously isn't an option, and the actual answer is $$48$$. Have I just interpreted the wording incorrectly, or is there a more fundamental mistake?

• $48$ is the correct answer.
– user655800
Commented Oct 3, 2019 at 16:59
• @HVxvejjw why not 49? Commented Oct 3, 2019 at 17:00
• On the first day there are $2^2=4$ mushrooms and the area covered by them is $1 m^2$ . On the second day there are $4^2=16$ mushrooms and the area covered by them is $9 m^2$. On the third day there are $6^2=36$ mushroom and the area covered by them is $25 m^2$ . So it will be on the $50th$ day my land will be fully covered by mushrooms. However note that we have started counting from the third day and not the first or second day. So the correct answer is$48$.
– user655800
Commented Oct 3, 2019 at 17:09

The First Term of the A.P should be taken as $$4$$ and not $$1$$ as $$1,4,6...$$ doesn't have common difference.
$$100 = 4 + (n-1)2$$ $$48 = n-1$$ $$n = 49$$
Hence the field would take $$49+1 = 50$$ days to be covered , but since Two days have already passed , required number of days is 48.
Count the number of squares formed by the mushrooms. Then we have the sequence $$1, 9,25,\dots,$$ or in other words, the squares of the odd integers, namely $$(2n-1)^2.$$ This also gives the area covered by the mushrooms. Since the field measures $$100×100,$$ it would have been covered when we have $$101×101$$ squares formed by the mushrooms.
This occurs on the $$k$$th day, so that $$2k-1=101,$$ or when $$k=51.$$ Indeed, it is clear from this why the answer cannot be an even number.