To solve the problem, we will define our variables, set up an equation, use inverse operations to find the solution, and finally present the answer clearly.
Step 1: Define Variables
Let \( C \) be the total number of candies Lucy collects after 7 days.
Step 2: Establish Patterns
- On the first day, Lucy collects 10 candies.
- Each subsequent day, she collects 2 more candies than the previous day.
Step 3: Calculate Candies Collected Each Day
- Day 1: 10 candies
- Day 2: \( 10 + 2 = 12 \) candies
- Day 3: \( 12 + 2 = 14 \) candies
- Day 4: \( 14 + 2 = 16 \) candies
- Day 5: \( 16 + 2 = 18 \) candies
- Day 6: \( 18 + 2 = 20 \) candies
- Day 7: \( 20 + 2 = 22 \) candies
Step 4: Using a General Formula
We can also express the daily collection as an arithmetic sequence:
- The amount she collects each day can be described as:
- Day \( n \): \( 10 + 2(n - 1) \)
Where \( n \) is the day number (1 through 7).
Step 5: Total Candies Collected
To find the total candies collected after 7 days, we can use the formula for the sum of an arithmetic sequence, where:
- The first term \( a = 10 \)
- The number of terms \( n = 7 \)
- The common difference \( d = 2 \)
The \( n \)-th term can be calculated using the formula: \[ a_n = a + (n-1)d \] So for the 7th day: \[ a_7 = 10 + (7 - 1) \times 2 = 10 + 12 = 22 \]
Now, we calculate the total sum of candies collected over 7 days: \[ C = \text{Sum} = \frac{n}{2} \times (a + a_n) \] Substitute the known values: \[ C = \frac{7}{2} \times (10 + 22) \] \[ C = \frac{7}{2} \times 32 \] \[ C = 7 \times 16 = 112 \]
Final Solution
After 7 days, Lucy will have collected a total of 112 candies.