SOLUTION: 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

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
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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