\begin{tabular}{l}\hline 109 Jim's Market couldn't keep Crunchy Critter Crackers in stock. Jim \\ started with 300 boxes but everyone wanted them. The first day Jim \\ sold 6 boxes, and on the second day he sold 14 boxes. Each day 8 \\ more boxes were sold than the day before. So after two days, he had \\ sold 20 boxes. If he kept selling the crackers at this rate, when would \\ Jim run out of Crunchy Critter Crackers? \end{tabular}

After initially selling 6 boxes on the first day and 14 boxes on the second -- a steady increase of 8 boxes each day -- we can establish the pattern of sales: 6, 14, 22, 30, and so on, following the formula \( 6 + 8(n-1) \) for day \( n \). To determine when Jim will run out of his initial stock of 300 boxes, we need to find out how many total boxes will be sold over a certain number of days. Let's calculate the total sales over \( n \) days: \[ \text{Total Sold} = 6 + 14 + 22 + \ldots + (6 + 8(n-1)) \] This can be simplified using the formula for the sum of an arithmetic series. The first term \( a = 6 \) and the last term on the \( n^{th} \) day is \( 6 + 8(n-1) \), which gives us a general term of \( 6 + 8(n-1) = 8n - 2 \). The formula for the sum \( S_n \) of the first \( n \) terms is given by: \[ S_n = \frac{n}{2} (a + l) \] where \( l \) is the last term. Now setting up the formula: \[ S_n = \frac{n}{2} (6 + (8n - 2)) = \frac{n}{2} (8n + 4) = 4n^2 + 2n \] To find when Jim runs out of stock, we set this equal to 300: \[ 4n^2 + 2n = 300 \] \[ 4n^2 + 2n - 300 = 0 \] Dividing by 2: \[ 2n^2 + n - 150 = 0 \] Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): \[ n = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 2 \cdot (-150)}}{2 \cdot 2} \] \[ n = \frac{-1 \pm \sqrt{1 + 1200}}{4} \] \[ n = \frac{-1 \pm 34.64}{4} \] Calculating the positive root: \[ n = \frac{33.64}{4} \approx 8.41 \] This means Jim will run out of boxes between the 8th and 9th day. Therefore, the 9th day is when he will completely sell out. To summarize: Jim will run out of Crunchy Critter Crackers on the 9th day, after a total of 300 boxes have been eagerly snapped up!