Using an arithmetic sequence, it is found that:
i) They collected 300 bottles on June 25th.
ii) The day was July 18th.
<h3>What is an arithmetic sequence?</h3>
In an arithmetic sequence, the difference between consecutive terms is always the same, called common difference d.
The nth term of an arithmetic sequence is given by:
![a_n = a_1 + (n - 1)d](https://tex.z-dn.net/?f=a_n%20%3D%20a_1%20%2B%20%28n%20-%201%29d)
In which
is the first term.
In this problem, the sequence is: {417 ,404 ,391, .........}, hence the first term is
, the common difference is d = -13, and the equation is:
![a_n = 417 - 13(n - 1)](https://tex.z-dn.net/?f=a_n%20%3D%20417%20-%2013%28n%20-%201%29)
Item i)
They started on June 15th, hence June 25th is the 10th day, thus:
![a_{10} = 417 - 13(9) = 300](https://tex.z-dn.net/?f=a_%7B10%7D%20%3D%20417%20-%2013%289%29%20%3D%20300)
They collected 300 bottles on June 25th.
Item ii)
This is n days after June 15th, for which
, hence:
![a_n = 417 - 13(n - 1)](https://tex.z-dn.net/?f=a_n%20%3D%20417%20-%2013%28n%20-%201%29)
![1 = 430 - 13n](https://tex.z-dn.net/?f=1%20%3D%20430%20-%2013n)
![13n = 429](https://tex.z-dn.net/?f=13n%20%3D%20429)
![n = 33](https://tex.z-dn.net/?f=n%20%3D%2033)
33 days after June 15th is July 18th.
More can be learned about arithmetic sequences at brainly.com/question/6561461