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Question 902479: A market had 300 cases of popular snack. The first day 6 cases were sold. The second day 14 cases were sold. Each day 8 more cases were sold than the day before. At that rate, how many days would it take to sell all the cases of the snack?
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! First we need the nth term of the arithmetic sequence (first term is a1 = 6, common difference is d = 8)
an = a1 + d(n-1)
an = 6 + 8(n-1)
an = 6 + 8n-8
an = 8n - 2
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Now plug S = 300, a1 = 6 and an = 8n - 2 into the formula below. This formula is used to compute arithmetic sums. The goal is to solve for n.
S = n*(a1 + an)/2
300 = n*(6 + 8n - 2)/2
300*2 = n*(6 + 8n - 2)
600 = n*(6 + 8n - 2)
600 = n*(8n + 4)
600 = 8n^2 + 4n
0 = 8n^2 + 4n - 600
8n^2 + 4n - 600 = 0
Solve the equation above using any method. You can use the quadratic formula, but I used a graphing calculator to get these approximate solutions
n = -8.913861725 or n = 8.413861725
Ignore the negative result
so focusing on n = 8.413861725, this means that we'll hit 300 cases somewhere between day 8 and day 9
Notice we have this table
| Day | Number Sold
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| 1 | 6 | | 2 | 14 | | 3 | 22 | | 4 | 30 | | 5 | 38 | | 6 | 46 | | 7 | 54 | | 8 | 62 | | 9 | 70 |
Now add up the values in column 2 from day 1 to day 8: 6+14+22+30+38+46+54+62 = 272
Now add up the values in column 2 from day 1 to day 9: 6+14+22+30+38+46+54+62+70 = 342
So that shows us the 300 target is somewhere between day 8 and day 9.
So to answer the question, it will take 9 days to sell out all of the cases.
Note: The answer is not 8 days because at the end of day 8, we have sold 272 cases and have 300-272 = 28 cases left over. That means only 28 cases will be sold on day 9 (instead of 70).
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