The Chocolate Dream store has just opened up and it is recording its sales. The store is open 7 days a week. In the first week, they sell 8 cakes each day. The next week, they sell 10 cakes each day. The following week they sell 12 cakes, and so on. During each successive week, they sell 2 cakes more than the days of the previous week. Based on this pattern, how many cakes will they sell in a 5-week period?
$8,10,12$
$\operatorname{arplit}$

Step 1 ：Given that the Chocolate Dream store sells 8 cakes in the first week and increases the number of cakes sold by 2 each week, we have an arithmetic progression with an initial term of 8 and a common difference of 2.

Step 2 ：We want to find the total number of cakes sold in a 5-week period. We can use the formula for the sum of an arithmetic progression: \(S_n = \frac{n}{2}(2a + (n-1)d)\), where \(S_n\) is the sum of the first n terms, n is the number of terms, a is the first term, and d is the common difference.

Step 3 ：In our case, we have \(n = 5\) (5 weeks), \(a = 8\) (8 cakes in the first week), and \(d = 2\) (2 more cakes each week).

Step 4 ：Plugging these values into the formula, we get \(S_5 = \frac{5}{2}(2 \times 8 + (5-1) \times 2)\).

Step 5 ：Simplifying the expression, we find that \(S_5 = \frac{5}{2}(16 + 8) = \frac{5}{2}(24) = 60\).

Step 6 ：\(\boxed{60}\) cakes will be sold in a 5-week period.