Etienne owns a small recycling company that picks up empty glass bottles from restaurants. At the first restaurant, he picks up 50 bottles. At each restaurant after this, he picks up 4 more bottles than he picked up at the restaurant before. Assume that this pattern continues.

a) Write the first four terms of the arithmetic sequence that represents the number of bottles he picks up at a restaurant. (4 marks)
b) Determine a and d for the sequence. (2 marks)
c) How many bottles will he have picked up after stopping at the eighth restaurant? (2 marks) (show your work)
d) If his truck can hold 2000 empty bottles, will he be able to pick up bottles at the 21st restaurant without emptying the truck first?

a) To find the first four terms of the arithmetic sequence representing the number of bottles Etienne picks up at a restaurant, we can observe the pattern provided.

First term: 50 bottles
Second term: 50 + 4 = 54 bottles
Third term: 54 + 4 = 58 bottles
Fourth term: 58 + 4 = 62 bottles

The first four terms of the arithmetic sequence are 50, 54, 58, and 62.

b) To determine the common difference (d) and the first term (a), we can use the information given.

Given that the first term is 50, we have a = 50.

To find the common difference (d), we can subtract any two consecutive terms.

d = 54 - 50 = 4

Therefore, the common difference (d) is 4.

c) To find the number of bottles Etienne will have picked up after stopping at the eighth restaurant, we can use the formula for the nth term of an arithmetic sequence:

nth term = a + (n - 1) * d

Given that n = 8, a = 50, and d = 4, we can substitute these values into the formula:

8th term = 50 + (8 - 1) * 4
= 50 + 7 * 4
= 50 + 28
= 78

Therefore, Etienne will have picked up 78 bottles after stopping at the eighth restaurant.

d) To determine if Etienne will be able to pick up bottles at the 21st restaurant without emptying the truck first, we need to find the total number of bottles he will have picked up by the 21st restaurant.

Using the formula for the nth term of an arithmetic sequence again:

21st term = a + (21 - 1) * d
= 50 + 20 * 4
= 50 + 80
= 130

Since the 21st term represents the number of bottles he would have picked up at the 21st restaurant, and it is less than the truck's capacity of 2000 bottles, he will be able to pick up bottles at the 21st restaurant without emptying the truck first.

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